On April 14, 2019, Cardinal Schönborn of the Archdiocese of Vienna gave a televised interview with the ORF (in German) with Gaby Konrad and Gerold Riedmann, where, among other topics, he was asked about and spoke about priestly celibacy. Taken together with various other statements he's made, his position seems to be, in effect, that it would probably be, all things considered, good to under certain circumstances ordain older married men to the priesthood in the Latin Church, but that the decision to do so should be a decision of the whole Church. Here is, in translation, that part of the interview.
Konrad: As you already mentioned, this Amazon Synod will take place in the fall, and then a second hot topic will be handled. The discipline of celibacy could possibly be relaxed. What is your take on the matter? Is the discipline of celibacy good, or is it bad?
Schönborn: The discipline of celibacy is what it is. I have experience of both, because I am also responsible for Eastern Catholic Churches in Austria, and I have around 30 priests who are married. I experience both sides: very good families, priests with families. And I experience very good priests who live celibately. And I experience crises on both sides: unmarried priests who do not maintain celibacy, and married priests whose marriages fail.
Therefore I would say, very soberly, life is not simple. But, if one lives the state of being unmarried so, as it was lived by Jesus as a model – the celibacy of priests is related to Jesus's having lived unmarried – that can go very well. It can also, unfortunately, as we see, go very wrong.
I am very convinced that one form is possible, and it will surely be discussed at the Amazon-Synod: the so-called "proven, married men."
In the Archdiocese of Vienna we have 180 deacons, most of them married, with a job, with a family. And I can well imagine that in the future there will not only be married deacons, but also married priests, who have proven themselves in their profession, in their family, and who are active in their communities, that such men can also receive the laying on of hands for priestly ordination.
Riedmann: That is to say, a testing similar to what you spoke of in connection with women in the Church. The deacons have… without the deacons life in the local church would probably no longer be possible in the same way.
Riedmann: To ordain these men as priests, is something you are in favor of.
Schönborn: Among the current deacons, married, with a job, so those who serve as deacons without a salary, there are certainly a great many, of whom one could say, they could perform priestly ministry very well. That is something that is being very earnestly discussed. It would of course create a two-class clerical system. The volunteer priests and the full-time priests. But why not?