$200 is for assembled boards. I learned electronics and spent about $200 on it in two years, that includes cheapest soldering iron. Don't order assembled boards when you are starting. Order cheapest bluepill (STM32F103C8T6) or non-original arduino clone and start on breadboard with that. Make pcb's only when you're ready to learn more. Expect that your first one will not work or will require some "rewiring", but second one may already work. You might start with some cheap through-hole components, they are a little easier to re-wire or re-solder, it's a good idea to put your first microcontroller in socket.

I'm a EE masters student and I also want to reinforce how inaccessible EE is and that I don't really recommend it as a hobby. EE is a very mature field and it's very math heavy for a reason. The second you move past the hobby boards, stuff becomes really difficult and really expensive really fast. If your end goal is to create toys for kids, then it's fine as a hobby. But without the formal training and lab access you are going to struggle to get past that point so it's pretty much impossible to turn the hobby into something more. Unlike software, where tinkering genuinely has the possibility of turning your side projects into careers. Hell, if you don't live in EE hotspot locations, I wouldn't even recommend it as a career anymore. Software is where it's at, even in the age of AI.

That's only for the heavy analog stuff. If you're into the more modern digital stuff, you basically never for any reason need to breadboard prototype, everything can be done with I2C modules and the like.

Burning a part is incredibly rare with this kind of stuff, if you're willing to put in the time to learn about it before actually building it.