Does there exist a chemical-free, homemade substitute to deli? Kids eat cold cuts cheerfully, whether alone or in sandwiches. I avoid salamis as their (saturated) fat content is conspicuous and alarming, but even the better ones, such as capocollo, are still filled with chemicals, with MSG, sodium erythorbate, and sodium phosphate present in those claiming to be superior because they contain no wheat.

The next best option is to prepare my own meat, slice it as thinly as I can, and use that. Unfortunately it'll be necessary to freeze and thaw, since otherwise the time investment would be prohibitive.

What is a class of homemade cold cuts that can replace store bought ones? I can think of roasting (slow cooking) lamb, pork, or chicken, suitably spiced with garlic, oil, and mustard, but the resulting flavor does not quite compete with the deli section. Does a healthy (chemical-free) homemade solution exist?
Roast beef is relatively easy to make at home and is much more delicious that store-bought.  Here is a good roast beef recipe from the New York Times.

There is a Japanese method of cooking chicken called "Torihamu" where the goal is to make the chicken the texture and flavor of deli ham.  Here is a little more information and a recipe.

A lot of the flavors of deli meat are dependent on salt.  A deli meat without salt just does not taste the same.  Remember that when making it at home, you will always be using less salt than a manufactured meat.  Celery salt may also give you the flavor you need.

There's also a lot of good recipes like this DIY Homemade Lunch Meat.

Here is a video for making deli ham.  The trick to that deli ham texture is cooking gently in water and then pressing.

I hope this was helpful!  Best of luck to you.