Delmonico's (in its mid-19th century incarnation) was the first a la carte restaurant in New York City, and insanely fashionable (it has its own steak cut named after it, for example). It's the kind of place that Dickens and Mark Twain had dinners with other famous people of their age.
It seems like the restaurant had a reputation for the best quality food, and it certainly invented a lot of dishes, like Baked Alaska, Eggs Benedict, Chicken a la Keene and Lobster Newburg. But a lot of those dishes seem incredibly heavy to us today, and a lot of it involves dowsing food in cream/lard/salt/sugar. And my understanding is that these meals would be washed down with large amounts of alcohol.
So - if someone were to make a period-correct Delmonico's dinner, just what would it taste like? Would it actually be appetizing to a modern palate? Would the supposedly fresh and high quality ingredients really be all that fresh and high quality in a modern understanding, or was the allure more eating incredibly rich, labor intensive foods? Or was it mostly the display/act of eating in a restaurant (with tablecloths!) with numerous waiters, bottles of wine, and elaborate table decorations like pièces montées?
Menus and recipes from Delmonico still exist, and recipes from around that time and before are also pretty readily-available, because Delmonico's most famous chef published a cookbook. This is a reasonably easy question to answer: you would definitely enjoy the food, but yeah, it'd be pretty heavy, and look a little different.
It's probably worth being a little bit clear here. Delmonico was around for a long time, and its menus were not carved in stone. What's popularly bandied about as its first menu is not actually from Delmonico; it's from another restaurant also named Delmonico's, so when you see, say, pork chops, pork and beans, or sausages and think "this is not fancy French food," that's why.
I'm going to talk a little more narrowly about the second half of the 19th century. Lorenzo Delmonico steps into the restaurant in 1831 and to a significant extent, makes it what it is. In this same time period, the Delmonicos had opened up a 220 acre farm in New York to get vegetables that weren't available to them, and a couple decades in, hires Charles Ranhofer as the chef de cuisine. Charles had the decency to write a cookbook that was published in 1894: "The Epicurean" which tells us just what the food was like and how to make it. It's available here if you'd like to study it: https://archive.org/details/epicureancomplet00ranhrich/page/416/mode/2up and it's well worth the investigation.
What was the food like under Ranhofer's guidance? As you'd figure, it was pretty heavy. It is also pretty labor-intensive, and it has an asethetic that's a little foreign to a modern eater. What's kind of amazing about it is how extensive it is: there are soups, sides, fish, (meat) entrees, vegetables, cold preparations, and deserts. What else is unique about it is that there is a significant reliance on wild game (wild duck, snipe, turtle) that you wouldn't find today because of dwindling populations, and because of changes in the law that would forbid selling non-seafood wild game.
The recipes work though, as you might expect. Grant Achatz is kind of a big deal - he's the chef (or I would I guess say 'restauranteur') at Alinea and Next. At Alinea, he made a recipe from Escoffier, circa 1906 Paris, on his first menu, which he described as an F-U (his words) to the culinary establishment. Why was it a middle finger? Because he was telling the world that he can do fancy modern stuff, but he can do old school too. Sort of like if a painter can bust out a copy of a Caravaggio without much effort - it's a show. And of course people loved it. In 2011, at Next (which basically 'reinvents itself' [or so it claims] every quarter or so), he did an entire Escoffier menu. People loved it too, because it's delicious.
Some of the recipes don't age super well. There's a lot of aspic, because jello hasn't been invented, so aspic was cool and difficult. There's a lot of cream and fat. There's a lot of consomme and other time-consuming stuff.
You also referenced seasonality. Here, if you look at the menus month-by-month, you'll see it play out. There are not a lot of green things on the menu in December and January. This makes sense because, as I've written elsewhere, cold storage containers and mass farming were still quite a ways off, so availability was limited. And Delmonico worked within those limitations. But the truth is, there are some, but not a lot, of salads on the menu in the warmer months, either, because the restaurants were giving people what they wanted.
Some sources:
Charles Ranhofer, The epicurean. A complete treatise of analytical and practical studies on the culinary art, including table and wine service, how to prepare and cook dishes, etc., and a selection of interesting bills of fare of Delmonico's from 1862 to 1894
Auguste Escoffier, Le Guide Culinaire.
Paul Freedman, Ten Restaurants that Changed America
Not precisely on point but still worth it for a time capsule into food is AJ Liebling, Between Meals: An Appetite for Paris
I have quickly scrolled through all the comments and it looks like none of them have mentioned that Delmonico's is still a functioning restaurant in NYC.
Not sure how much the menu had changed, but I watched a recent video about them and it looks like they take the history of the place very seriously. I think they still prepare some some of the original recipes.