
I would most certainly disagree with that. I have come across quite a few priests who disagree with basic Catholic teaching on a range of issues (female ordination, the validity of Anglican orders, homosexual relationships etc.) and then there is the issue that while some priests are indeed very learned, some others (although they may be good pastoral men) aren’t exactly the sharpest tools in the box as regards intelligence, and you would entrust them with such a decision as determining whether God had originally joined a couple in marriage in the first place.
This would seem to be a way in which people can get the outcome the wish simply by finding a parish priest who is ‘liberal’ enough to grant them an annulment. This would seem not to be based around carefully determining whether God had joined the two people in marriage in the first place, but around making it easy for people to get the ‘result’ they want.
If a priest grants an annulment to a marriage that was actually valid in the first place, then despite having had the original marriage signed off as invalid, any subsequent marriage is not actually a marriage in the eyes of God.
Handing the decision on whether to annul a marriage over to parish priests would in reality make a Catholic annulment an easier thing to obtain than a civil divorce, and without having to involve anyone trained in law of any sort (Canon Law or otherwise). That would be a travesty and a mockery of Christ’s words on marriage.