Or maybe a better question, can you keep playing the same game after Diablo dies? Seems like you cant to me.

With singleplayer character, no, the game ends when you defeat Diablo, but with a multiplayer you can continue.

You can continue on a new cycle of the game in Singleplayer, ending your current playthrough. Anything you left lying around anywhere vanishes. There's nothing you can loot from Diablo himself.

In multiplayer, everybody who was on level 16 is kicked from the game, served to the outro and earning a kill on that difficulty setting (a dot on your battle.net icon if that's where you're playing; one dot for each difficulty). If you're in town or in a different level, you'll be left to continue playing in that session, but don't count as getting the kill on Diablo. Diablo doesn't drop anything in multiplayer, either.

That's not quite true. The Dark Lord is not a boss monster, but it is a regular monster. Regular monsters can drop things, but are not guaranteed to drop things. For drop purposes, The Dark Lord is mlvl 30 (Diablo), or 45 (Hellfire), before being modified for difficulty. This means that in Diablo, its drop is of the same probability and quality as a Blood Knight or Advocate, both of which are far more numerous and easier to kill. In Hellfire, the boosted mlvl may allow The Dark Lord to drop better items.

If you disable the ending movie with Diablosaver you can loot diablo... safe download link and instructions:

https://d1legit.com/2019/04/06/diablo-1-diablo-saver-character-back-up-scan-others-for-dupes-hacks-and-more/

Another method is to cast firewall on him repetitively and then telekinesis. Cast the spell and wait for him to die, if he drops anything you can pick it up with the telekinesis spell you have already queued.

save the game when he starts bleeding and then load, now you can walk around and pick up the loot,