I hate to say it, but since it doesn’t boot I don’t believe there is a way to make it work without some sort of repair to the connector. If the circuit board itself is ok, then someone skilled with hot air rework tools could probably put a new connector on (you might find a friendly local small assembly shop willing to try).
The JTAG connector in theory could be used, but you’d need to be someone inside of NVIDIA to use it, and the JTAG hardware itself is far more expensive than the board. JTAG is less likely to work than is a skilled hot air rework bench tech (you could buy a lot of Xaviers for the price of the JTAG hardware…and then you wouldn’t be able to use it without sending to NVIDIA).
I don’t have any particularly simple ways to get around this. There was a recent addition to the TX2 line (normally 8GB of RAM), and the module of this sells for the same price as a TX1 module…unfortunately, this doesn’t come with a carrier board, so in reality, unless you already have a carrier board for a TX2 the bare module wouldn’t help much. On the other hand, there is an educational discount for a TX2 dev kit. The carrier footprint is larger and there would be significant work transitioning from 32-bit to 64-bit, but performance would go up substantially. Don’t know what your options are, but it is something to consider.