My understanding is that both Finder searches and TimeMachine are dependent on Spotlight. Spotlight lasted less than a day on Sequoia after two search attempts I decided it was totally useless. Spotlight is missing two very important abilities. One is scheduled indexing, the other is limiting the metadata index to only the files you want to search.
That said I do not use TimeMachine and have kept Spotlight disabled all the way back to Tiger. When I need to do searches I use FindAnyFile. Highly recommended! If you recall OS-X Panther, FAF does searches in exactly the same way and yes it's fast. And it will search locations that Spotlight will not.
NOTE: System updates will probably re-enable Stoplight so you'll have to disable it after the update along with turning off Apple Intelligence, Wifi.....