been an XiiaLive Pro supporter for years, recently found a new free app (PlayTime Radio) and the devs said they are going to add encrypted recording very soon. I have both installed, don't get me wrong. XiiaLive because I'm used to the interface and has my favorite stations collected over the years, so I use it to listen to stations. PlayTime for discovering new songs as they're more song-centric (and has a beautiful and simple interface).
There are also apps like Audials and Internet Radio Recorder but not sure if they'll get into legal trouble/blocked as they export as .mp3 files with totally no protection.
also, can confirm TuneIn encrypts and saves their files (in the folder /data/data/radiotime.player/cache/(some numbers) if anybody wants to check it out) and they don't seem to be blocked by many radio stations, if any. Totally regret buying TuneIn now that it's become bloated and greedy though.
Jona: personally would prefer recording over android auto and chromecast actually, but it's your app (and you're one of the best devs ever) so will respect your decisions.
Customer support service by UserEcho