I’ll take a stab at the potential cause of the problem... noting that this information is based entirely upon observed behaviour.
As emails are downloaded to the Mail inbox, they are locally “cached” on the device - and during this process are indexed by the system. When email messages are subsequently moved to a folder, the “index” (locally stored on the device) is updated. When searching for an email, it is the index that is actively searched - not the email messages as might be anticipated.
Assuming that you are using an IMAP mailbox, all email messages are held on the server until they are deleted - at which time deleted email is fully deleted from the server. When deleted, your iPad sees the deletion(s) and updates its local index.
If you switch to a new device, when setting-up email, only the recent existing email message headers are downloaded from the IMAP mail server - not the body text. As such, the local index only contains new Mail messages and the message headers of recent email.
You will likely see this mechanism at work if your start by looking at your most recently received email - where you will instantly see both the header and body text. Now, move down your inbox, one message at a time; again you should see both header and body of recent messages - these are all indexed and searchable.
Now, scroll down through your inbox; you will very likely see your Mail client briefly pause as it retrieves a “block” of Mail headers (but not body text) from the Mail server. When accessing these messages, only then is the body retrieved from the Server - and is locally indexed - to make it searchable on the device.
When switching between devices, I have noticed that it has been necessary to individually access each email message (in each folder) to rebuild a searchable local index on the device.
This seems to be a fundamental limitation of iOS/iPadOS, to which only the described (time consuming) workaround seemingly exists. There doesn’t appear to be a mechanism that actively downloads an entire Mailbox, from the Server, when setting-up an email account on a new/replacement device.
Whilst not being able to provide a solution, I hope this insight provides some useful explanation of the potential issue.
