Nobody can realistically maintain bases in a country without some sort of agreement with the local government (and a certain level of tolerance from the population at large) or an expensive full-on occupation. As far as I know, there is a single US base on a territory where the local government does not want it (Guantanamo, Cuba), literally on the doorstep - anywhere else would be prohibitive to maintain long-term hostile occupation.

Everything else is maintained and operated in agreement with local authorities - which is why the US, at the moment, cannot use Spanish bases and Diego Garcia to wage war on Iran. Even Saudi bases have been blocked in the past (notably to invade Iraq).

Without long-term bases, it becomes extremely difficult to project power with continuity. Can you still do the occasional special op, like killing Osama? Sure, but you can't do things like ensuring free navigation (and hence the flow of resources and goods) and signal intelligence gets so much harder.