Your image of the charging port shows bent pins and possible missing ones as well. B1, B4, B9 are bent and B5, B8, B10 and B12 seems to be missing (broken off)
Here's a link that shows the pinout for a USB-C port that may help as it explains the functions of the pins.
You will have to replace the USB-C port.
As it is hard mounted on the motherboard you will need smd soldering skills and tools.
Search online for Lenovo ThinkPad X1 Carbon 6th Gen USB-C charge port (supplier example only) to find suppliers that suit you best.
Here's a link to the hardware maintenance manual that will also help as it details how to remove the systemboard which is necessary if you wish to replace the charge port.
if it still doesn't show that it is charging after the port has been replaced, what is the model number of the motherboard as printed on the board itself? The schematics will have to be found to find out what else was wrong
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