I've also read that those with ordinary mission must accept those as having an extraordinary mission (ie. not just miracles). Extraordinary mission isn't determined by those who are not those with ordinary mission.  That would mean acceptance is necessary by the current Novus Ordo hierarchy.  Unless the Traditional bishops can show they actually are the ones with ordinary mission.

Here is part of it:

I say, in the second place, that never must an extraordinary mission be received when disowned by the ordinary authority which is the Church of Our Lord. For (1.) we are obliged to obey our ordinary pastors under pain of being heathens and publicans (Matt. xviii. 17): - how then can we place ourselves under other discipline than theirs? Extraordinaries would come in vain, since we should be obliged to refuse to listen to them, in the case that they were, as I have said, disowned by the ordinaries. (II.) God is not the author of dissention, but of union and peace (I Cor. xiv. 33), principally amongst his disciples and Church ministers; as Our Lord clearly shows in the holy prayer he made to his Father in the last days of His mortal life. (John xvii.)

How then should he authorise two sorts of pastors, the one extraordinary, the other ordinary? As to the ordinary- it certainly is authorised, and as to the extraordinary we are supposing it to be; there would then be two different churches, which is contrary to the Most pure word of Our Lord, who has but one sole spouse, one sole dove, one sole perfect one (Cant. vi.) And how could that be a united flock which should be led by two shepherds, unknown to each other, into different pastures, with different calls and folds, and each of them expecting to have the whole. Thus would it be with the Church under a variety of pastors ordinary and extraordinary, dragged hither and thither into various sects. Or is Our Lord divided (I Cor. i. 13) either in himself or in his body, which is the Church?-no, in good truth. On the contrary, there is but one Lord, who has composed his mystic body with a goodly variety of members, a body compacted and fitly joined together by what every joint supplieth, according to the operation in the measure of every part (Eph. iv. 16).

Therefore to try to make in the Church this division of ordinary and extraordinary members is to ruin and destroy it. We must then return to what we said, that an extraordinary vocation is never legitimate where it is disapproved of by the ordinary.

We must then return to what we said, that an extraordinary vocation is never legitimate where it is disapproved of by the ordinary.

I don't see that as the opposite.  The last sentence you highlighted summarizes what he has been saying. He is stating that the ordinary must approve.  Otherwise, it is illegitimate.

The issue is....the NO hierarchy are supposedly the "ordinaries" with ordinary mission.  They certainly do not accept any of the Traditional groups as having extraordinary mission. 

(3.) And in effect where will you ever show me a legitimate extraordinary vocation which has not been received by the ordinary authority. S. Paul was extraordinarily called -but was he not approved and authorised by the ordinary once and again? (Acts ix. 13). And the Mission received from the ordinary authority is called a mission by the Holy Spirit (ibid. xiii. 4.). The Mission of S John Baptist cannot properly be called extraordinary because he taught nothing contrary to the Mosaic Church, and because he was of the priestly race. All the same, his doctrine being unusual was approved by the ordinary teaching Office of the Jєωιѕн Church in the high embassy which was sent to him by the priests and Levites (John i. 19), the tenor of which implies the great esteem and reputation in which he was with them; and the very Pharisees who were seated an the chair of Moses,- did they not come to communicate in his baptism quite openly and unhesitatingly? This truly was to receive his mission in good earnest. Did not Our Lord, who was the Master, will to be received by Simeon, who was a priest, as appears from his blessing Our Lady and Joseph; by Zachary the priest; and by S. John? And for his passion, which was the principal fulfilment of his Mission,-did he not will to have the prophetic testimony of him who was High Priest at that time.

Fr Hesse reduces the factual doubts about the new rite to personal doubt - "I never had any personal doubts about my ordination." 

A personal doubt is called a 'negative doubt' because it's based on feelings.  A positive doubt is based on facts/evidence.  His personal doubts are irrelevant to the matter because no one can ever know with exact certainty.

Fr Hesse at the 56 min mark then makes more logical errors.  He says "The Church, in doubt, is usually in favor of validity."  ....but...then he quotes canon law which says "Only if you have POSITIVE and REASONABLE doubt is validity not presumed."

Ding, ding, ding!!!  We have a winner. 

There is positive/reasonable doubts about the V2 rites, thus, as Fr Hesse says, canon law does NOT presume validity.

These rites directly contradict Pius XII's infallible decree on the necessary form/prayer, in order to be valid.  This is positive/reasonable doubt.

I strongly recommend you read Fr Calderon's study on the NREC, which hopefully will be posted soon, to educate yourself better on this subject.