There are often controllers which do indeed just mimic the signals. Doesn't work with every appliance, it depends on the way it's implement and if the manufacturer wanted to make that approach infeasible.

But there absolutely are options to record such Signals and then replicate them via home assistant - I used them before to control a ceiling fan and various infrared devices (same idea, but not a radio there instead a "blaster" - I think it was called)

I didn't set it up again after my last move though, as I couldn't mount the ceiling fan in this apartment and the Infrared devices were just my media center (tv, audio), which are hardly in use currently