What can cause narrow disc brake gap in TRP Spyre-C?
That's an uncommon problem for mechanical disc calipers and I have a guess what's going on.

Spyres and Spykes are generally good brakes but they have the quirk that the pad adjusters can get a little stuck. When this happens, the adjuster will feel seized, but if you're only trying to turn it out, that could be mistaken for it being retracted all the way. Try turning it both ways to confirm it's moving freely.

If it is stuck, what's always worked for me is take a higher-leverage 3mm wrench in very good condition (to minimize the risk of rounding the wrench flats) and just pop it free. It can take more force than you'd normally want or expect to be putting on a 3mm allen, and when it frees it makes a scary 'pop,' but then works normally again. 

If both pad adjusters are known to be moving freely and all the way retracted and the cable isn't even attached and the arm is known to be all the way in its starting position, that kind of only leaves some kind of internal problem with the brake, and I don't know what that would be.