Oh great, the new rite of ordination was said in Latin, wonderful. This is no different than the New Mass in Latin, the new rite of episcopal consecration in Latin, or for that matter let us read the New Code of Canon Law in Latin. The words in the new rites do not signify what they ought to signify. The sacramental words do not signify what they once signified under the older traditional rites.
When I was a seminarian under Bp. Williamson, he insisted that Fr. Brian Hαrrιson be conditionally ordained. Fr. hαɾɾιson was ordained by Card. Cody with the new rite; and Cody was a valid bishop. I do not see that Latin makes any difference here.
And if what Newbie says is true, that Fr. Hesse received conditional ordination one year before his death, this makes things more bizarre. This means that Fr. Hesse entertained some form of doubt. I am not sure that the story is true though.
I don't know Bryan, in the early 70s, a few times we went to the NO mass that was done ad orientem in Latin and entirely "by the book," by an old priest ordained in the 30s, and from the pews you could not tell any difference between that and the TLM. I mean if you did not know, you would think he was saying the TLM. IIRC, I think the prayers at the foot of the altar and the confiteors were omitted, those were the changes that you noticed from the pews.
So it was likely the same with his priestly ordination, except if something was changed or omitted that invalidated his ordination, certainly at some point he would have known or discovered it.
I don't know about the nun who said he was conditionally ordained, because Fr. Hesse said he never doubted his priestly validity, yet he still put the matter to +ABL and the others who knew very well what was invalid and valid. It also makes no sense whatsoever that he would want her to tell no one.
For me, I would say that if a man like Fr. Hesse says his ordination was certainly valid, then I would say you can be sure that he did all the homework.