Do they even make a 3" closet flange?
Yes. Lots of them.
Possibly the most common is a "4 in. x 3 in. " Closet Flange. It glues either outside of a 3 inch pipe or inside a 4 inch pipe.
There are 3-inch inside. 3-inch outside. The 3-inch inside are often avoided because they have the narrowest passage, although that passage is still bigger than the output of a toilet. There are all kinds of variations.
A 4 inch pipe or closet bend has the advantage that both the inside and outside can be used over time, which can come in handy during rework/remodel.
Some flanges use compression rather than glue. Those would seem to be an advantage to those who are less practiced and might be concerned about the need to get it right the first time with glue. I am not a plumber.
Stainless steel rings on the closet flange are more sturdy than plastic. They are more rust-resistant than painted steel, although the painted steel should be good for decades.