In that recording, Fr. Hesse talked about his ordination, he said:
"...I have been ordained, unfortunately in the new rite of ordination, but thank God in Latin, everything strictly by the book and +ABL said that would be valid, +Fellay said it's valid and Fr. Franz Schmidberger who is my present superior in Austria says it's valid and +Williamson said there's no need for conditional ordination..."
Hesse was ordained by a Traditional Catholic bishop. So that would not be an issue.
But he was ordained using the New Rite of Ordination. The New Rite contains a different Sacramental Theology of the Priesthood.
The Traditional Rite asks God to "bestow these instruments [the tools]," specifically the tools to offer the "salutary sacrifices," which are the consecrated hands, paten and chalice. The New Rite speaks ambiguously about "sacrifices of the Tabernacle" that were "a shadow of future goods" and which are replaced by Jesus when he "offered Himself." The New Rite says nothing about the "instruments" [tools] necessary to perform the Sacrifice.
In the New Rite, the Bishop says that the Old Testament Sacrificing Priest [Sacerdotes] was a mere metaphorical "figure," a shadow, which was replaced by Jesus, once and for all, when he died on the Cross. That is the Protestant view. Then the bishop asks God to give him "a helper" to carry out his work, whatever that may be.
In the Traditional Rite, the bishop asks God to give the new Priest the power to offer the Holy Sacrifice just as the OT Priests offered sacrificial victims for sins.
The different prayers ask for different things. You get what you ask for and no more. As Leo XIII said in
Apostolicae Curae, "All know that the Sacraments of the New Law, as sensible and efficient signs of invisible grace, ought both to signify the grace which they effect, and effect the grace which they signify." Therefore, the New Rite is, at the very least, positively doubtful in conferring the Sacerdotal Priesthood on ordinands.
TRADITONAL PRIESTLY ORDINATION PREFACE
Vere dignum et justum est, aequum et salutare, nos tibi semper, et ubique gratias agere, Domine sancte, Pater omnipotens, aeterne Deus, honorum auctor et distributo omnium dignitatum ; per quem proficiunt universa, per quem cuncta firmantur, amplificatis semper in melius naturae rationalis incrementis, per ordinem congrua ratione dispositum. Unde et Sacerdotales gradus, atque officia Levitarum, Sacramentis mysticis instituta creverunt : ut cuм Pontifices summos regenverunt : ut cuм Pontifices summos regendis populis praefecisses, ad eorum societatis et operis adjumentum, sequentis ordinis viros et secundae dignitatis eligeres. Sic in eremo per septuaginta virorum prudentium mentes Moysi spiritum propagasti ; quibus ille adjutoribus usus, in populo innumeras multitudines facile gubernavit. Sic et in Eleazarum et Ithamarum filios Aaron paternae plenitudinis abundantism transfudisti ; ut ad hostias salutares, et frequentioris officii Sacramenta, ministerium sufficeret Sacerdotum. Hac providentia, Domine, Apostolis Filii tui Doctores fidei comites addidisti, quibus illi orbem totum secundis praedicationibus impleverunt. Quapropter infirmitati quoque nostrae, Domine quaesumus, haec adjumenta largire ; qui quanto fragiliores sumus, tanto his pluribus indigemus. Da, quaesumus, omnipotens Pater, in hos famulum tuum Presbyterii dignitatem ; innova in visceribus eorum spiritum sanctitatis, ut acceptum a Te, Deus, secundi meriti munus obtineant, censuramque morum exemplo suae conversationis insinuent. Sint providi cooperatores ordinis nostri ; eluceat in eis totius forma justitiae, ut bonam rationem dispensationis sibi creditae reddituri, aeternae beatitudinis praemia consequantur.
It is truly worthy and just, fair and salutary, that we should always and everywhere give thanks to you, holy Lord, almighty Father, eternal God, author of honors and distributer of all dignities; by whom all things progress, by whom all things are established, always being enlarged for the better by the growths of the rational nature, arranged in order according to a suitable reason. Whence also the priestly degrees, and the offices of the Levites, were instituted by the mystical sacraments, as when the supreme pontiffs ruled: as when the supreme pontiffs presided over the governing peoples, for the aid of their society and work, you chose men of the next order and of second rank. Thus in the desert you propagated the spirit of Moses through the minds of seventy wise men; with whom he used auxiliaries, and easily governed innumerable multitudes of the people.
Thus you poured out the abundance of the paternal fullness of Aaron's sons, Eleazar and Ithamar; so that the service of the priest was sufficient for the salutary sacrifices, and for the more frequent office of the Sacraments. By this providence, O Lord, you added to the apostles of your Son teachers of the faith, with whom they filled the whole world with their second preachings.
Wherefore we beseech thee, O Lord, to bestow these instruments on our infirmity; the more fragile we are, the more we need these. Grant, we beseech you, almighty Father, to these your servants the dignity of the Presbytery; renew in their bowels the spirit of sanctity, that they may receive from Thee, O God, the office of second merit, and insinuate censure by the example of their conduct. Let the co-workers of our order be provided; let the whole form of justice shine forth in them, that they may return the good account of the dispensation entrusted to them, and obtain the rewards of eternal happiness. (Google Translate)
NEW PRIESTLY ORDINATION PREFACE
Adesto, Domine, sancte Pater, omnipotens æterne Deus, humanæ dignitatis auctor et distributor omnium gratiarum, per quem proficiunt universa, per quem cuncta firmantur, qui ad efformandum populum sacerdotalem ministros Christi Filii tui, virtute Spiritus Sancti, in eodem diversis ordinibus disponis. Iam in priore Testamento officia sacramentis mysticis instituta creverunt : ut cuм Moysen et Aaron regendo et sanctificando populo præfecisses, ad eorum societatis et operis adiumentum sequentis ordinis et dignitatis viros eligeres. Sic in eremo, per septuaginta virorum prudentium mentes Moysi spiritum propagasti; quibus ille adiutoribus usus populum tuum facilius gubernavit. Sic in filios Aaron paternæ plenitudinis abundantiam transfudisti, ut ad sacrificia tabernaculi, quæ umbra erant futurorum bonorum, meritum sufficeret secundum Legem sacerdotum. Novissime vero, Pater sancte, Filium tuum in mundum misisti, Apostolum et Pontificem confessionis nostræ Iesum. Ipse tibi per Spiritum Sanctum semetipsum obtulit immaculatum, et Apostolos suos, sanctificatos in veritate, missionis suæ participes effecit; quibus comites addidisti ad opus salutis per totum mundum nuntiandum atque exercendum. Nunc etiam infirmitati nostræ, Domine, quæsumus, hos adiutores largire quibus in apostolico sacerdotio fungendo indigemus. Da, quæsumus, omnipotens Pater, in hos famulos tuos presbyterii dignitatem; innova in visceribus eorum Spiritum sanctitatis; acceptum a te, Deus, secundi meriti munus obtineant, censuramque morum exemplo suæ conversationis insinuent. Sint probi cooperatores Ordinis nostri, ut verba Evangelii, eorum prædicatione in cordibus hominum, Sancti Spiritus gratia, fructificent et usque ad extremum terræ perveniant. Sint nobiscuм fideles dispensatores mysteriorum tuorum, ut populus tuus per lavacrum regenerationis innovetur et de altari tuo reficiatur, utque reconcilientur peccatores et subleventur infirmi. Sint nobis iuncti, Domine, ad tuam deprecandam misericordiam pro populo ipsis commisso atque pro universo mundo. Sic nationum plenitudo, in Christo congregata, in unum populum tuum, in Regno tuo consummandum, convertatur.
Be present, O Lord, holy Father, eternally almighty God, author of human dignity and dispenser of all graces, through whom all things prosper, through whom all things are established, who, by the power of the Holy Spirit, arrange in the same different orders to form a priestly people, the ministers of Christ your Son. Already in the former Testament the offices of the mystical sacraments were established: that when you presided over the people by ruling and sanctifying Moses and Aaron, you chose men of the next order and dignity to assist their company and work. Thus in the desert, through the minds of seventy wise men, you propagated the spirit of Moses; by the help of which he governed your people more easily.
In this way you poured out the abundance of the father's fullness on the sons of Aaron, so that for the sacrifices of the tabernacle, which were a shadow of future goods, the merit was sufficient according to the Law of the priests.
Last but not least, holy Father, you sent your Son into the world, the Apostle and Pontiff of our confession, Jesus. He himself offered himself to you through the Holy Spirit immaculate, and he made his apostles, sanctified in truth, sharers in his mission; with whom you added companions to preach and carry out the work of salvation throughout the whole world. Now, even in our weakness,
we ask, Lord, to grant us these helpers whom we need in our apostolic priesthood. Grant, we beseech you, almighty Father, to these servants of yours the dignity of the priesthood; renew in their bowels the Spirit of holiness; let them obtain the office of the second merit received from thee, O God, and insinuate the censure of their manners by the example of their conduct. May they be honest collaborators of our Order, so that the words of the Gospel, by their preaching in the hearts of men, by the grace of the Holy Spirit, may be fruitful and may reach the ends of the earth. May they be with us faithful stewards of your mysteries, so that your people may be renewed through the bath of regeneration and restored from your altar, and that sinners may be reconciled and the weak may be relieved. May they join us, O Lord, in imploring your mercy for the people entrusted to them and for the whole world. Thus may the fullness of the nations, gathered in Christ, be converted into one people of yours, to be consummated in your Kingdom. (Google Translate)