This PR is an alternative to #35509. Rather than fixing, it removes the --perf option / profile_with_perf context manager features (both introduced in #14519) from the functional test framework. Given that no developer apparently even noticed that it's been broken for more than a year, we can conclude that it's largely unused and getting rid of it to reduce maintenance burden seems a reasonable choice. So far, neither in #35509 nor on IRC anyone has signaled strong interest to use or wanting to keep this feature.
The corresponding test documentation is removed as well, though the mentioning of perf profiling is still kept in the general developer docs -- only the reference to functional test framework integration is removed there.