My old MBPs motherboard has a faulty HDD connector, are there any other places on the motherboard I could connect storage to?
No - the HDD interface is SATA/AHCI and not USB. I suppose if you were really skilled, you could somehow solder on to the USB inside or re-route one of the connectors / ports.

Have you looked at the extremely tiny USB sticks that barely take much space outside the connector?

Something like SanDisk Cruzer Fit or similar low profile model: https://www.amazon.com/SanDisk-Cruzer-Low-Profile-Drive-SDCZ33-016G-B35/dp/B005FYNSZA