Your biggest problem is sharing a console with someone else. That's not a good way to be and that's also the main reason you're out of HDD space. Other than that, just play game Y while game X is installing, or do something completely different.
People usually have the whole 500GB or more to themselves and I only play one game at a time anyways. Maybe that's part of the problem, too. You have too many games going on simultaneously, and your attention is divided, so you just kind of tune out instead of following through with one story at a time and really getting into it. Instead of experiencing a journey from start to finish, you go in intermittently like "where was I again, what was this about again?".
I have 1Tb to myself and pretty much only play GTS. However I have a bandwidth cap. Deleting old games also deletes all the multi GB updates. If I wat to play them again it will take a long time to get all the updates again. Plus most my PSVR games are digital downloads, deleting them feels like throwing games away :/ I know I can re-download them, but bandwidth cap.
For example, I still have DriveClub on there. I have the disc yet that's so little of the game that it will take hours to download all the updates if I want to do a quick race again.