I like the Eastern Orthodox approach. If a bishop lays hands on a woman (to ordain her), she receives nothing and he is no longer a bishop. The East believes that the bishop has excommunicated himself in doing a deed contrary to God's will."Grunt. East good. West bad. Make me feel better about not being Catholic."
That's a distinct difference between the Eastern and Western approaches to ordination. In the West we tend to think of Holy Orders as adhering to the man ordained, who then, even if excommunicated, is a valid priest or bishop and whose acts, though perhaps illicit, are "valid." The East, on the other hand, tends to see ordination as belonging to the church and as valid only in the context of full communion with the Church. Thus the bishop ordaining a woman would, in the West, still be a valid bishop, while the one doing so in the East, by incurring excommunication, renders his ordination ineffective or void, and ceases to be a bishop. There are gradations and variations in these approaches but this is, in outline, the direction each takes.
I have always felt that the western understanding is mechanical, and it seems to assume that the Holy Spirit is compelled to act regardless of the orthodoxy (or even membership in the Church) of the priest. If an apostate or heretical priest, even one excommunicated by the Church, can consecrate the Eucharist, ordain, or absolve sin, then he is, in effect, able to make the Holy Spirit do his will. I do understand the historical reasons Augustine came up with this understanding, but I think he was mistaken and that Cyprian was correct on the matter. I think Lombard takes the same line as you when it comes to the Eucharist. He often seems more in accord with the Eastern Fathers than do most western writers.
I like the emphasis on the church first; it cuts out the byproduct of our Western Catholic sacramentology, the many independent bishops (vagantes). Rightly understood of course it's Catholic, as you can say about the Christian East generally. The problems here are you can fall for the heresy of thinking the validity of the sacraments depends on the worthiness of the minister, and the Orthodox are so narrow they don't really think we're even baptized, let alone ordained. With them, divorce and remarriage and now contraception are okay but we're not baptized. Go figure. To me they sound like they're idolizing their own tribe. No, thanks. So I'm glad the Catholic Church errs on the side of mercy, even if it means annoying vagantes junking up the Internet. What's really funny, given the real Orthodox' narrow sacramentology depending on being in the church, is when vagantes, who think "Orthodox" just means "high church without the Pope," claim they're Orthodox.
Those who have apostolic succession, such as Catholics and Orthodox, tend not to pull out charts to prove their "lines" of it.
I like our quasi-branch theory, not a divided church but that our criteria for valid orders, orthodoxy so basic the Nestorians pass, unbroken line of succession, and unbroken true teaching about the Eucharist (sacrifice made present and complete change of the elements, so the Anglicans are out), both take into account that every ancient church including ours claims it is the only true one yet the others are still part of the great Catholic family. Related: the first seven councils (Orthodox defined doctrine) and the Vincentian canon more or less give you Catholicism. Uncanny.
Good old Augustinian, Western Catholic sacramentology is why the Patriarch of Moscow is Msgr. Kirill to us (a bishop but without jurisdiction), not Mr. Gundyayev (not a bishop).
It is nonsense to say the Protestants although holding many heresies do not have the Holy Spirit and valid baptism. The Holy Spirit seems a lot less worried about heresy than we are!Sounds Pope Francissy of me but I agree.