I've been playing Dave the Diver and it might be exactly what you're looking for, 1 dive is 15-20 minutes with loads of progression and a fun story.
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I think Battle Bit is a good one for this kind of things even though it is not single player that you are looking for. In this game, you could just hop in one of the 127vs127 server, shoot like a maniac and have fun and maybe revive a few crews and so forth. As you rack up points and kills, you unlock stuff for your gears and that is some of the progressions you feel earned as you play the game. You could just hop in for one session and leave anytime without having to make any commitment.
In Battle Bit, it's basically anything goes, it's chaotic and fun, it doesn't feel like a competition either that you sometime see in PVP matches on other games.
A few thoughts:
You could try roguelikes (not modern roguelites) like ADOM or DCSS. While when you die you start over you will have learned a lot and a beginners winning run is likely to take about 12 hours of dungeon diving (it saves after every action so you can quit and resume whenever) in DCSS for example. They're free, and DCSS is constantly updated so you can really play that game on and off for a lifetime if you like. Because they're so lightweight they also open and close almost instantly, which can be a plus if you're pressed for time. Hell you can play DCSS from anything with an internet connection and a screen using the online server!
Adventure-puzzle games like myst, obduction, talos principle? Long breaks between sessions could lead to you forgetting things, but short regular playtime is how we solved this style of game as kids. No loot but clear progress.
Minecraft remains ever popular?
You could look into emulating some old handheld games on your phone? or grab something like a modded 3DS and use that. A lot of handheld games had pretty pick up and play approaches, and with emulators you can just suspend the state whenever you have to stop even if there's no save point which fixes a lot of the problems with older games. I can recommend giving chronotrigger a shot on an emulator!
Rogue Legacy
Final Fantasy 1-6
umm, under 10GB?
Anyway, Single player: Death Stranding. Multiplayer: Dead by Daylight