I think that depends on the restaurant. There used to be a small chain around here with a BBQ/southern menu; their bread pudding definitely was not dry.
It surely depends on the restaurant, but the dry variant seems to be the most common - probably because if not sold in volume, you can make a larger batch and reheat for sales as needed, while staying visably ‘fresh’ for longer. The more custard-styles (like I make) become less ideal much, much faster even in the fridge.
And yes, as @Saint_Cad pointed out earlier, you can make low quality/overly dry/shelf stable variants and cover with a ‘high’ end sauce. I recall visiting several stores where when you see bread pudding on the menu they mention (sometimes) the source bread ‘brioche bread pudding’ and then 2 sentences about the sauce ‘topped with our famous -Insert brand name- bourbon, Hawaiian cane sugar, locally sourced butter, scraped Madagascar vanilla bean, and the finest Ceylon cinnamon.’. When you see that sort of description in a menu, you should have no surprises. 
Any restaurant offering southern cuisine should be able to make a good bread pudding without overdoing the sauce. If they can’t, they need to consider changing their menu.
That’s a good point. Lots of them really have been forgotten, though, and I think it’s less a matter of convenience mixes and more a matter of people forgoing even mixes in favor of ready-made items. I haven’t seen gingerbread mix or pineapple upside-down cake mix in ages. I don’t see ready-made gingerbread (not cookies but gingerbread, the kind you eat with applesauce or hard sauce) in bakery sections, either. They’re just gone.
My mother was an immigrant, as were many of my friends’ moms when I was growing up. None of them gave up their native dishes, but they did supplement them with American dishes and with convenience foods. A lot of the women who’d exhausted themselves cooking from scratch for threshers and/or large families happily embraced anything that would get them out of the kitchen faster, especially if those items (processed and chemically as they were) promised better nutrition.
I collect vintage cookbooks. I have Arlene Francis’s no time for cooking cookbook that features vacuum-sealed meats. Here’s an excerpt:
Many a Twentieth Century homemaker gets home just before dinner and starts coking before taking off her coat and hat…Let’s face it: as housewives, most of us are miracle-happy. We’re the luckiest cooks in world history…Many of our goods are so miraculously processed, pre-prepared and packaged that miracle is the only word to use for them. For our convenience, ease, and well-being, we have three adoring, full-time godfathers: the scientist, the dietitian, and the manufacturer.
Space-age, processed heaven! 
I have a Whip’n Chill cookbook that has recipes for prune whip and Charlotte Russe. All three are gone–Whip’n Chill for good reason.
Going to disagree in the specific and agree in the general. For the specifics, Gingerbread mix is still very much a thing, and I see it in the stores, although yeah, there’s 5-10 different kinds of yellow or chocolate cake to each box of it. And Pineapple upside down cake is normally made with a can of sliced pineapples and a box of yellow cake mix plus some minor tweaks.
The ready made stuff is made to sell to the widest market - thus I normally only see the stuff that turns over pretty fast, and therefore is either really cheap, or really unremarkable. How many boxes of chocolate chip cookies will I see in a store bakery? Dozens. How many sugar cookies? At least a handful. How many peanut butter or oatmeal cookies? Mabye one or two. Anything else, not so much. Same with donuts, same with pretty much any product. More rare desserts have less market, and thus less market space, leaving only the specialties and home cooks. Don’t see that changing.
I don’t think I’ve ever baked anything using a commercial mix (although I’ve eaten them, and they usually taste like chemicals to me), but I imagine that things like gingerbread mix go in and out of style.
Pineapple upside down cake was famously invented via a contest by Dole Pineapple for canned pineapple recipes. It was a very 1950’s food item. There was a fad then for a kind of decorated, molded dessert, usually using canned fruit. I have a recipe book from then that was full of inedible-looking things like a a plate of canned fruit made into a face (peach half was the head, cottage cheese was the hair and beard … ).
Good to know. It’s not in either grocery store here within walking distance of me. I’ll have to check at Christmas time, when people may make gingerbread cookies. (For the record, I never buy it. I’ve got a killer recipe that’s pretty easy.) As for pineapple upside-down cake, of course most people make it from cake mix and a can of pineapple. 'Twas ever so. My mom made it that way. But you used to be able to get a box containing cake mix, pineapple, and a packet of brown sugar mixture because I guess buying that stuff separately was less convenient?
Oh, my gosh, that’s hysterical. Was it called a “salad”? I ask because I have the old Betty Crocker Cookbook for Boys and Girls that features salads like Raggedy Ann Salad (peach half body, hard-boiled egg head, grated cheese hair) and Bunny Salad (canned pear half body, almond ears, cottage cheese tail). And then there were all those Jello “salads”. The worst one was lemon Jello with shredded carrots in it.
Sorry, I digress.
Ha! Me too. Every birthday I get German chocolate cake. I require nothing else. That is my gift and all I need.
I’ve seen it mentioned, but I’ll add another echo for bread pudding. I absolutely love a good bread pudding.
I suspect that you’ve never seen the horror that is a Jello mold with meat encased in it?
That hot dog thing looks like the winner of a “I can make the worst aspic ever” contest.
*shudder*
What if we add Spaghetti-Os?
It’s the Vienna Sausages that really do it for me. To which group of party-goers would you serve that?!? 
There was something about gelatin that people couldn’t get enough of.
Sort of like kale is now.
There are only two legitimate uses for kale: Colcannon and decoration on salad bars. 
So no molded raspberry kale salad with for you, then?
Way too many prepackaged salads contain kale. Example:
https://scontent.harristeeter.com/legacy/productimagesroot/DJ/7/1470247.jpg
My SO loves the stuff but I find it overwhelms everything else. Bleh!
I misread this as “salad bras.” I mean, I guess kale would be a legitimate use there. Better than cauliflower, that’s for sure.
Bread pudding is basically reformatted French Toast - so it makes eminent sense 
Jell-o in general has fallen out of fashion. When we were growing up, Jello with a dollop of Cool Whip was considered a real treat.
I suppose hospitals still serve it, but I’d be frightened to try it.
I’ll have a little cup of Jell-o with chunks of fruit in it for a snack, but I wouldn’t serve it for dessert. And if I top it, I use real whipped cream. Cool Whip is nothing but aerated lard and sugar, and I never did figure out what went into Whip ‘n’ Chill. Until today, I had forgotten it even existed!