Oddly enough, I actually have taken my kids to several of those:
—Took them to Avec on Super Bowl night a few years back. Wouldn't on a normal night. Just barely enough things they'd be willing to eat, and of course the crowd on a normal night would be a pain.
—Great Lake, sure, a couple of times. But always early, when we were first customers of the night.
—Katsu: did this just because a relative was in town and really wanted to try this place I'd raved about. We did okay, but generally a frustrating experience, simply because of the cuisine.
—A Tavola: we took Myles to this before Liam was born but it proved to be so hushed that I wouldn't have dared with two kids. He was very good, but I wouldn't recommend it and it's one where I'd genuinely feel my kids' presence was unfair to other diners seeking a grownup place.
In general, Mexican can always be made to work though it's hardly worth it at the higher end places (I'd wind up spending a lot of money at a Geno Bahena place to get three little masa boats which they'd scrape the good stuff off of); Italian always works; middle eastern is generally fine, they both like falafel and schwarma; they'll do Indian though it's possible the younger one will subsist entirely on naan; Chinese can be made to work. Other Asian gets harder, I've pretty much given up on Thai with them, too, because it's so hard to find something that fits their definition of acceptable food.