Mentzer's BECMI codified the ultimate D&D endgame: apotheosis. The execution might leave something to be desired, but I think the basic idea is a good one. Instead of Immortals waiting to welcome newly ascended adventurers into their pantheon, it might be cooler if they treated godhood as something they didn't particularly want to share. It's got to be taken from them.
The gods are probably too absorbed in their own activities to spend a lot of time actively going after adventures. Probably. But they're certainly not going to make the paths to immortality easy to find, and likely going to put obstacles in the way of adventurers who go after them. The more powerful they get, the more they'll attract the Immortals' attention and be bedeviled by them. Think the sort of things that happened to Hercules and others hunted by the gods in Greek myth.
Immortals as adversaries or obstacles would certainly explain some of the things about dungeons and other adventuring locales. The only problematic detail would be clerical magic. I suppose clerics, empowered by the Immortals, might eventually become adversaries to other adventurers. They would be sort of the gods' check to make sure humans didn't get too powerful. That would be interesting, but maybe too game-changing. Alternatively, clerics might be powered by the fundamental forces of the universe (the same thing that powers the Immortals) and militant humanists bucking the gods by using that power for the good of mankind.