AMPLIFi remote is basically a gearbox style software and you can use it while recording. It works fine as long as you don’t switch apps frequently and set the phone to do not disturb.
It has a visual interface where you tap and select amps and pedals visually, it’s quite nice. Also you can stream music and backing tracks to jam out to on it under the music tab. Plug into your PC and use your headphones or external powered monitors and you’ve got a nice practice amp setup. Additionally I’d recommend downloading Audacity, it’s a free DAW that’s extremely easy to use and would let you record your own tracks and let you edit audio to learn songs as well (slow down the tempo, change keys, etc) and the process is actually pretty easy.
Yes, the cloud for amplify is not the same as custom tone servers, that is why it isn't working. You will have to reprogram that tone on your apple device in Amplifi Remote.