Revamp installed_on_request handling #31228

Job	Run time
syntax	51s
update-test (Ubuntu)	24s
cask audit	2m 53s
update-test (macOS)	29s
tap syntax	6m 18s
formula audit	4m 39s
vendored gems	50s
test default formula (Ubuntu 22.04)	1m 28s
tests (online)	3m 6s
test default formula (Ubuntu 20.04)	3m 30s
tests (generic OS)	2m 52s
test default formula (macOS 13 x86_64)	3m 33s
tests (Ubuntu 24.04)	3m 7s
test default formula (macOS 15 arm64)	1m 15s
tests (Ubuntu 22.04)	3m 7s
tests (Ubuntu 20.04)	3m 8s
tests (macOS 13 x86_64)	3m 7s
tests (macOS 15 arm64)	3m 8s
	47m 45s

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