After clicking the button which indicates that I have completed my portion of the trade, a popup asks if the site should also remove those cards from my inventory and tradelist. I clicked ok as per usual, but those cards were still present.

Thanks for reporting this. I have just tried and it seems for me it works. Are you sure those cards were present in your inventory with the same details (if there was no red background highlight in the trade listing, then they were present correctly, and they should have been removed).

I think that was behavior I saw during the site transition concerning a trade that was started before the change. I just (like just now) performed the same procedure for a trade set up after the site change and my inventory and tradelist do seem to have been modified correctly.