Self-rising flour isn't particularly weird or chemical-y. It's just all-purpose flour, baking powder, and salt. I never buy it, but it's super-common here in the US South, and I guess in parts of Australia and New Zealand as well-- a lot of recipes from those regions specify NOT self-rising when they mean regular flour. There's also self-rising pastry and self-rising whole wheat flour. Since it's so common regionally, sometimes it's actually fresher.
And that biscuit-style pizza crust is very typically Midwestern. I don't think I had "real" yeast-risen pizza until I was in high school. It can be really tasty comfort food if you know what to expect. Chef Boyardee and a bunch of other brands make "kits" with that type of crust, sugary sauce, and powdered cheese.
It's white flour, so it's not ideal, but it's SO much better than takeout or a kit. It's a good transition, and if you work at it, gradually you can make it more healthy-- slowly swap out the self-rising flour for whole wheat with a little baking powder, and gradually add more veggies and less cheese and meat. Eventually you'll develop something that's pretty good for you.
(You might also want to look into recipes for no-knead refrigerator yeast dough. You can use pretty much any whole grain flour with yeast, water, and a little salt. You mix it and let it rise once, then store it in the fridge for up to two weeks. When you take it out, it has to rest but not rise-- letting it sit while you prepare the toppings is usually enough. It takes prior planning, but on the day you actually make the pizza, it's just as fast as the self-rising but lets you avoid the white flour.)
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