Arent battle royale games inspired by things like The Hunger Games or Battle Royale? All the server minigames like that that I recall from back in the days were named something like Hunger Games

Yes. The Hunger Games film and book inspired by the Japanese film "Battle Royale" in turn, inspired the Minecraft minigame. But later battle royale games were inspired by the minigame, not the films directly. A shrinking world border, for instance, is pretty important to make the concept work (in a film, it doesn't actually have to work!).

Last man standing formats were perfectly possible in traditional FPS formats too, but they weren't really a thing because to actually be fun, the format needs

1. Big maps and lots of players (more than the typical FPS)

2. A "searching for loot" mechanic, where you can increase your chances of survival by looking for good items, making interesting risk/reward tradeoffs and discouraging just turtling up in the most defensible location.

3. Shrinking borders, to prevent an anticlimactic endgame of powerful players searching for hiding stragglers.

Minecraft basically had all three since 2014, and there were quite popular last man standing formats like UHC even before they had world border (and before the Hunger Games film came out).