Yes. Combine your approach with a new device, a mailbox size of several gigabytes and EDGE/2G network connection. IMAP/JMAP will work fine in those conditions,

Yes. Combine your approach with a new device, a mailbox size of several gigabytes and EDGE/2G network connection. IMAP/JMAP will work fine in those conditions, as a client can simply fetch the list of folders and a few email headers, and then lazily fetch email bodies on demand. Updating the read status is pretty cheap and does not require a full sync between the device and server, and so on.