QuickTime screen recording has no sound

That's expected, unfortunately. QuickTime can record the microphone but macOS doesn't let it grab the Mac's own sound. You need a virtual audio device. BlackHole is the free one people use. Install it, then in Audio MIDI Setup make a Multi-Output Device with your speakers and BlackHole both ticked, and set that as the Mac's output so you can still hear things. In QuickTime, click the arrow next to record and choose BlackHole as the mic. Set the output back to your speakers afterwards, volume keys don't work on a multi-output device.