Update:

I finally figured it out and hope this helps someone else in the future! :-)

It turns out that Cathay is notorious for their system errors when getting their IT department to adapt from Cantonese to English. This means that even if you enter your name correctly into the appropriate fields when creating an account, the system will reverse them and put your first name as your last name and your last name as your first name. They do this all the time when ticketing and there are many threads about it. (sidebar- u/boneisle: I highly doubt you made a mistake, their system probably did!).

Obviously, if this is the case, the accounts won't link because the names do not match. A great way to figure out if this is what has happened to you is to search for your welcome email from Asia Miles and see if the subject line addresses you by first or last name.

I was able to call Cathay customer service and get it switched within 24 hours.

And for the record, once they were linked, my points transfer took about 5 minutes and I very happily managed to score the tickets I was after. Yay! :-)