Revamp installed_on_request handling #31290

Job	Run time
syntax	39s
tap syntax	6m 7s
cask audit	2m 55s
formula audit	4m 57s
tests (online)	4m 56s
update-test (Ubuntu)	23s
tests (generic OS)	3m 5s
update-test (macOS)	28s
tests (Ubuntu 24.04)	3m 19s
vendored gems	40s
tests (Ubuntu 22.04)	3m 5s
tests (Ubuntu 20.04)	4m 6s
tests (macOS 13 x86_64)	8m 22s
tests (macOS 15 arm64)	5m 5s
test default formula (Ubuntu 22.04)	1m 27s
test default formula (Ubuntu 20.04)	2m 53s
test default formula (macOS 13 x86_64)	4m 9s
test default formula (macOS 15 arm64)	1m 41s
	58m 17s

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