Far too kind

The client can certainly work around it, yes. But it's a pain in the ass design. For example, if you want to wait for new mail to appear in multiple mailboxes, you have to make a bunch of connections to the same server, one per mailbox, just so that each one can sit idle, watching for new mail in the one single mailbox. And some clients try to do this. Others give up push notification, and poll for new mail periodically.

Back to the problem with SEARCH: To implement multi-mailbox search, you need to invoke the SELECT command and the SEARCH command once per mailbox. (remember: this is all on a single server!) With my 44 mailboxes, that'd requires 44 SELECT/SEARCH commands. You could parallelize it by using multiple connections, but that's still a bunch of extra work. I find it unlikely that you'll be able to make that competitive in speed with a single search on your mail client's fulltext index.