Well, you're technically correct (the best kind). They work with first gen VTXes when they're upgraded to the V01.01 firmware. The catch is that if you upgrade your V1/V2 Goggles will no longer work with those VTXes.. And now there are Goggles 3 that aren't even compatible with O3 VTX yet, and it's 50/50 on whether or not they'll work with first gen - it's all a pretty big mess compatibility wise :)
DJI remotes work with Caddx Vista as well (the predecessor to O3); those are digital video systems that offer much better quality than analog.
I have no experience with BetaFPV remotes, but I know they're cheap...