How do I replace the water supply line that appears to be built in to the shutoff valve?
I've dealt with these before. The flex pipe is silver soldered into the valve body, probably a product design of the day to dispense with having a problematic compression fitting that would eventually leak.

If you loosen the nut from the pipe nipple, the valve body and flex supply line pipe will be rotatable to position it for a direct lineup to the faucet inlet.



Even if the other end of the metal flex supply line is compatible with the faucet (sounds like it isn't), brass and copper flex work harden from bending into position and can crack if disturbed for later repairs, so I don't recommend reusing it. The only way to fix this is to replace the shutoff valve with a 1/2" to 3/8" right angle shutoff. The newer ones available are nice quarter turn units. Remove the compression fitting nut and ferrule and the 3/8" supply line should thread right on. They use a neoprene flat washer now that gets rid of the leak problem the original silver-soldered supply line was meant to prevent.