If you would like for your own posts to be reachable by the search system, all you have to do is go to Preferences > Public Profile > Privacy and Reach. Once there, check the box that says Include Public Posts in Search Engines.
I’m happy that this is opt-in for users, though something that would be good to know is how users on other instances (as well as platforms like Lemmy, Kbin, Misskey, etc.) are handled. I guess the search system would not handle these users?
I get that searching can be a bit finicky sometimes but doesn’t typing in a full username of a user you want to search for usually do the job?
That part about shutting down is something that https://joinmastodon.org/covenant tries to help with, where advance notice should be given and multiple people should have access to administrative actions. At least if the server has to shut down the users are given enough time to look at another server.