Are you suggesting that if I want to search on a particular device, then that device should hold my backup storage? If so, then I disagree and have chosen to store my backups on a device that's behind two locked doors, hard to reach, well protected from being accidentally disturbed, and which doesn't run unrelated software.
Okay, suppose that I were to back up all of my mail to each client device; three at the moment, or four if you count my old phone, which is still running because I haven't gotten around to wiping it. Then I'd have five backup sites instead of one. But four of those are backed up to the fifth, so the size of the mail backup would be quintupled, the activity on the backup device would increase and with it the number of possible occasions for a failure, and the age of my oldest backup would be automatically decreased to compensate.
