You know, there's no requirement that your gift come before or at the wedding.
You say you usually prefer to buy a "thing gift" and send it to the couple. So go ahead and do that anyway and let it be delivered after the wedding. Not a thing wrong with that.
And if you can't find registry information, so what? Choose something nice. Fight the creeping notion that guests are only supposed to choose gifts that the couple themselves already selected -- it's simply not true. I wince every time I read the expression "off registry gift" as if it's some sort of odd exception to the norm.
I'm with you in general. Few people dislike the 'creeping notions' surrounding the Wedding Industry more than I do. In this case, however, I had never met anyone involved in the wedding before. Not the bride and groom; not the bride's parents... nobody. We were invited by the Bride's dad, who works with my husband. And they aren't particularly close friends, having only worked together for 2-3 months... So I would have had a very hard time choosing a gift for the couple. I could have sent a gift after the wedding (I did - on the day of the wedding - find out where they were registered); but my husband comes from a bring-a-gift-to-the-wedding culture and he would have been very uncomfortable showing up empty handed - more uncomfortable than I was going with a cash gift.
Anyway, I took cash in a card and kept it in my purse until I could see what the gift table setup was. If it had seems unsecure, I would have given the card to the couple or her father directly. As it was, the card box was provided by the reception venue and was very secure -- a locked metal filligree box with a slot in it. The box itself was secured to the table. So I was able to stick the card in there and not think twice.
Lovely wedding, BTW.