I think there are essentially two ways of making the compiler statically aware of lifetimes: you can annotate the program with lifetimes, as in Rust, or you can have the compiler infer all of the lifetime information somehow.
Adding annotations requires more programmer effort and reduces the level of abstraction of your programming language. Rust takes this approach, but tries to keep the level of annotation somewhat reasonable, and provides the "unsafe" escape hatch for when you cannot convince the compiler that what you're doing makes sense. (Although I do believe that it would be possible for the Rust compiler to infer most/all lifetimes).
For instance, in a functional programming language, you are typically creating closures all over the place, like one of the other posters mentioned. It seems quite unlikely that there is any kind of reasonable lifetime annotation system that would capture all of the complex lifetime patterns that you would see in such a language.
One can say "Oh, but surely someone smarter than me must be able to come up with something that works for those situations", but I don't think that is a very constructive line of thought, since you could say that about pretty much everything. Rust is an example of a language where lifetimes work reasonably well, but Rust already makes some tradeoffs that reduce its level of abstraction to some degree. It doesn't provide the same level of abstraction that, say, Haskell does.
Program analysis would allow you to keep your level of abstraction and ideally not require only little annotations. However, since any non-trivial program property is undecidable, program analysis is fundamentally incomplete and it is again unlikely that it would be able to deal with all lifetime patterns that you see in practice.
Therefore, it seems to me that there is no free lunch here. Either you sacrifice abstraction by requiring annotations or you don't require any annotations but then you have an incomplete program analysis.
(Also, you seem to think 30 years of good research has been wasted on the "obviously bad idea" of garbage collection, but somehow none of these researchers were good enough to realize this and come up with the silver bullet that makes GC unnecessary? After all, people were already thinking about GC-less languages long before Rust was a thing)