I can understand why a customer doesn't want to open a brand new unit. (fear to damage something else or not just not being confident to do it)...This is perfectly understandable.
What I do not really understand, is that you told GHL that you didn't wanted to open the unit and you wanted them to fix the issue and that you didn't had problems with the downtime.
Then just after sending the unit you ask for a refund

...Supposedly because you don't trust anymore on the unit due a led not turning on ?
To answer your question "What kind of connector came loose during transport?"
I purchased 2 brand new MP40QD , one of them came with Wifi not working.
I logged a ticket to Ecotech and same as GHL they asked me to open the controller and check the Wifi chipset. what I have done and it comes that the chipset was loose, I just pushed it back in and the Wifi was working.
Another one with Calfeeder of Pacific Sun reactor with a loose screw inside of the controller. Connector can come loose during transport YES.
I faced some issue with GHL products and both Matthias G/F were all time there for me and standing behind their products with great customer care.