Note that these aren’t trivial to implement correctly and performantly based on existing searchsorted functions.
Would interface like this make sense in Base or in a package? Or maybe it already exists somewhere?

The first example seems challenging with binary search (since the function is not monotone, hence, it is hard to find the starting point and the binary search might overlook the relevant range).

For monotone functions you could use LazyArrays.jl. If the function is monotone decreasing, than one would need to add rev=true. For example

That’s pretty clever!
I already use mapped arrays (a much lighter mapview instead of BroadcastArray) for searchsorted in a different context - when searching by f(A[i]) == x instead of A[i] == x:

using SplitApplyCombine
searchsorted(mapview(f, A), x)

However, for <, >, ... searches, the interface in the first post would be simpler and more consistent than searchsorted(mapview(>(5), A), 1). A couple of issues with the latter:

doesn’t support == predicate

the rev argument doesn’t correspond to how the array is sorted: its meaning is different between sort(A; rev) and searchsorted(mapview(<(5), A), 1; rev) leading to potential confusion

Not extensible: the predicate interface I outlined also allows for stuff like searchsorted(∈(1:5), A) and searchsorted(∈(1..5), A). With the current searchsorted interface, the IntervalSets package had to define a separate function searchsorted_interval(A, 1..5).

not intuitive, while searchsorted(>(5), A) is basically a more efficient drop-in replacement for findall(>(5), A) when the array is sorted.

Oh, and the mapview/BroadcastArray solution can be significantly slower. For A = 1:10000, x = 1234 they are 25 ns instead of 3 ns for searchsortedfirst($A, $x).