20 or so minutes cooking the meatballs in the sauce has always done the job for me. You want to cook them sufficiently long so that they absorb the moisture and flavour of the sauce, but not too long such that they lose their firm texture. The reverse process of the meat flavouring the sauce is also an important one.
As a side note, it is more advisable to simmer the sauce, rather than boil it, or it will likely become too thick and may even burn.
In general, the combination of initially cooking the meatballs in the oven, and then simmering in the pan is the right way to do things. This ensures that the meatballs are sufficiently cooked, do not fall apart, and have the right firmness. Worth noting is that you should pad off the grease on the meatballs before putting them in the sauce.
I think you've got a wide berth of possibilities but what's above seems pretty restricted. What you are listing above sounds like a pretty straight-forward Midwestern American chili. It has the features of the standard chili spices, beans and tomatoes, with a nice variety of meats.
As @Cold suggests, beets would be great to add for their sugar content. However, there are quite a few American recipes that utilize potatoes, so you might seek those out. Are there other vegetables to substitute that you recall as more native? Other beans maybe? How about caraway, dill, chervil, tarragon?
Vodka would seem like an easy route to go to appease the indigenous aspect, but I don't think it would yield much results in terms of impacting the flavor; especially in the context of the extant chili recipe. Unfortunately, I can't see the benefit of buffeting vodka against the flavors of cumin, coriander, clove, let alone hot peppers. On the other hand, Russians produce some amazing beers (I am a big fan of Baltika), and I would recommend looking into switching to a lager flavor or Imperial Stout as I mentioned above.
But for real, my spin would be to approach this from a Solyanka point of view (which would definitely give you ceiling room to try incorporating Medovukha); or really any of the other amazing cold Russian soups. Or perhaps try to incorporate mini dumplings like Pelmani, mini Kotlety, or use Shashlyk-style prepared meats for the chili. I would also consider trying to get some lamb in the recipe, in particular at the expense of the chicken.
But don't forget, chili is all about what you want to put into it (<-- self-promotional plug); not what the standard template lists.
Best Answer
Getting meatballs done is a matter of raising them to the correct internal temperature. 30 minutes seems like a long time even for large meatballs, but it depends on many factors. The best way is to pull the biggest one out and test it, first see if it's firm or squishy - firm means it's getting close to done. If it's firm stick an instant read thermometer into the very center of it, if the temperature is at or above 160F (70C) then they are all done. If you test a medium size one the larger ones may need a little longer.
Incidentally, boiling meatballs is not the only way to do it, and my least favorite because some of the flavor is lost to the water. You can bake them, fry them, or cook them in a sauce as well. Cooking them in a sauce imparts flavor to the sauce, you'll get the fat from the meat too, though.