They do, XBox One and ps4 anyway. Only WiiU still plays from disc.
Even if the games were on cartridge they would still install to the HDD, simply to be identical to the download version. Saves testing 2 different versions. Also we already have 8GB patches, how big does a cartridge need to be ensure all patches will fit on it as well. Consoles could really benefit from a faster HDD (and controller) I rather have them invest in building games for M.2 SSD drives next gen, than wasting money on cartridges. Games are still going to be installed regardless.
It would be nice if you could back up patches and savegames on an ssd card to store with your discs for easy re-install later. Retrieve patches from SSD should be an option. Simply add a little prompt while deleting a game, store patches on SSD. While installing, ask retrieve patches from SSD.

I know games install stuff on my PS4, but I don't think I can play any of my games without the disc can I?

No. So you want a code with the disc that makes the disc useless after install? It won't be like the old PC discs you can install again on a different PC.
If only people were honest and not abuse any system where you could simply play the game without the disc, then we wouldn't need all this drm crap. Any convenience factor (like the ability to play a digital version on multiple consoles) is heavily weighed against how much abuse it could get.

I'm not sure which one would be considered the better one now (since I'm hearing cartridges may be becoming viable again) but I prefer cartridges over discs.

True of course, it's not so easy anymore, but yeah I would like it like old PC discs.

Discs also have cases to store them in:

From the point where disks begin to require installs, cartridges win. Let PS5 be catridge based please.

Carts, although "cartridges" isn't really accurate to say anymore since games would most likely come on flash-type memory cards as opposed to actual cartridges like the NES/SNES/N64 which had circuit boards, ROM chips and batteries for backup and were much more expensive to manufacture.

Nintendo could have 32GB or 64GB type cards mass-manufactured pretty cheaply, and the combination of no physical disc drive plus being able to store additional save/game data on the cards themselves as opposed to a large capacity HDD built into the console would also save on the manufacturing costs.

On top of that assuming the NX turns out to be a hybrid console / handheld device, being able to play the same game cards on both devices would be awesome.

The picture you posted is not showing up to me for some reason. Anyways, I know that, but discs are big and most of the time putting them in multi-disc cases will scratch them. Heck, I took great care of my Wii games and they still had scratches on them, it's pretty annoying. Can't wait for optical media to die, it had a long run =P

It doesn't really matter to me, but it will be interesting to see the NX use carts!