Do I have to run a full Bitcoin node?No, LND supports
neutrino; c-lightning allows using a pruned node. Eclair users are out of luck. Mobile clients obviously do not need you to run a Bitcoin node.
I have no idea if this was discussed yet since this thread is damn long, but I would like to mention that I tried c-lightning with a pruned node, since it was mentioned everywhere that it should be possible, then ran into issues. When asking in the community, I was told that indeed those issues were due to me running Bitcoin Core pruned, so I had to go back to square 1 and restart Bitcoin Core in non-pruned mode.
Also PSA for newbies: Raspberry Pi Model 3B+ with a (fast) USB 3.0
HDD takes literal
weeks to sync and verify the whole blockchain. Either have a real desktop (maybe also laptop with enough umph) Linux PC with which you can do the initial sync, then plug the drive into the Pi. Or use an SSD. Just wanting to prevent tons of headache

Especially since in case of an issue with the HDD (give it external power!!), when Bitcoin Core has to restart, it has to re-verify a ton of blocks which can easily add a day or two for each crash that you experience.