I guess these emails are hidden. You can try MFCMAPI to open the PST file on the new computer and check if they exists or not. With the raw properties of these emails, you can get to know whether they exists or not.

Also in your last test " I've tried on the old computer to Export the Sent Items folder to a new Pst and then copy the file to the new computer and the sent items folder is empty.". What is the size of the new PST file containing these 10, 000 sent emails. If it is very small, then the emails are not exported successfully.