I don't think you're missing something
simple. Recording streaming-audio raises copyright issues, so most operating systems I'm familiar with don't have built-in ways to record your system audio.
I'm on macOS and there is a product called
BlackHole that can do this, but it requires some setup. Based on your OS you might want to search for ways to record your system audio. Once set up, you should be able to record system audio directly in AmpliTube 5 or a Digital Audio Workstation.
If you do it in a DAW, once you have your play-along track recorded, drag that audio track into AmpliTube's recorder on a different track. Turn off AmpliTube for that track. You don't want your practice track getting processed by an AT chain.
You would only arm your guitar track for recording. Then hit the Record button and play along. Then you could hit play to hear them together or export the song. Remember that AmpliTube only records your raw guitar and sends it to your current chain for processing. It doesn't capture any real-time MIDI data, so if you are using any MIDI pedals to make changes in AT, those won't get captured. I've found it's better to use a full-featured DAW once your needs get beyond just recording your raw guitar.