pageserver: use tokio::time::timeout where possible (#4756)

Removes a bunch of cases which used `tokio::select` to emulate the
`tokio::time::timeout` function. I've done an additional review on the
cancellation safety of these futures, all of them seem to be
cancellation safe (not that `select!` allows non-cancellation-safe
futures, but as we touch them, such a review makes sense).

Furthermore, I correct a few mentions of a non-existent
`tokio::timeout!` macro in the docs to the `tokio::time::timeout`
function.

docs/pageserver-thread-mgmt.md

@@ -30,8 +30,8 @@ or similar, to wake up on shutdown.
 
 In async Rust, futures can be "cancelled" at any await point, by
 dropping the Future. For example, `tokio::select!` returns as soon as
-one of the Futures returns, and drops the others. `tokio::timeout!` is
-another example. In the Rust ecosystem, some functions are
+one of the Futures returns, and drops the others. `tokio::time::timeout`
+is another example. In the Rust ecosystem, some functions are
 cancellation-safe, meaning they can be safely dropped without
 side-effects, while others are not. See documentation of
 `tokio::select!` for examples.
@@ -42,9 +42,9 @@ function that you call cannot be assumed to be async
 cancellation-safe, and must be polled to completion.
 
 The downside of non-cancellation safe code is that you have to be very
-careful when using `tokio::select!`, `tokio::timeout!`, and other such
-functions that can cause a Future to be dropped. They can only be used
-with functions that are explicitly documented to be cancellation-safe,
+careful when using `tokio::select!`, `tokio::time::timeout`, and other
+such functions that can cause a Future to be dropped. They can only be
+used with functions that are explicitly documented to be cancellation-safe,
 or you need to spawn a separate task to shield from the cancellation.
 
 At the entry points to the code, we also take care to poll futures to