I also definitely would not eat there. I don’t know why it’s assumed that you would eat dinner there after being stood up. I’d be sad and I’d want to be alone, at absolute most I’d get takeout, but there’s probably a greasier takeout that I want, but have been restraining myself from. I’d very much prefer tacos/falafel to a date restaurant meal if I’ve just been stood up.
I’m your age and I’ve done it several times, including with my husband.
The caveat is that you start with coffee or a drink (my husband and I arranged to play mtg and have a beer), then the conversation is so nice that you order food or move from a cafe to a restaurant.
Now that I think about it, all the good relationships I’ve been in that weren’t with friends of mine involved dinner tacked onto the first date. When I’ve dated friends, it’s a very different progression, but doesn’t really involve restaurant dates at the beginning.