Mail just sometimes refuses to issue a server-side search via IMAP. It insists that it wants to search its offline database of downloaded mails. When it cannot find anything, it will even tell you that “Mail downloads and organizes messages when iPhone is locked, charging, and connect to Wi-Fi.” And my response to that is, it has had eight hours to do exactly that every night.

That's fair then (though I haven't hit it because I have all my mail downloaded). There's other bits in Mail that are flaky with Gmail compared to standard IMAP mail providers, though.