The Mail app on iOS & iPadOS does not gracefully handle IMAP mailboxes that contain a large number of messages. In my usage, the "magic number" seems to be around 10k, but YMMV. The issue manifests in various ways: in a large "Inbox," new messages may not appear on the iPhone. A large "Sent" or "Archive" mailbox may result in the iPhone not storing messages to that mailbox, e.g. emails sent from the iPhone appear in the "Sent" mailbox on the iPhone but not on the IMAP server. The Mail app never generates an error message to the effect that it can't actually store or retrieve messages, or sync the mailbox. It just fails silently.
The workaround I have used successfully is to sort the messages on the server into multiple archive folders (by year is an easy choice) to reduce the per-folder message count to something that the Mail app can handle.
The workaround I have used successfully is to sort the messages on the server into multiple archive folders (by year is an easy choice) to reduce the per-folder message count to something that the Mail app can handle.