Signed or unsigned distance in electric potential due to point charge(s)
I would assume that the potential should be positive whenever our (unit)test-charge and the point-charge repel each other, and negative when they attract each other.

This assumption is true for system of two charges of which one of them is +ve test charge but if there are more charges in the system you can still apply that logic but taking two charges at a time but not for system of charges altogether and for that you need to know about superposition theorem.
In Superposition theorem(in this case) ,you can remove all the charges from system of charges  except test and one charge(Q1) from the system  and calculate potential due to that one remaining charge (and you can apply attraction-repulsion logic here because there is only two charges .
And repeat above  process again but with test charge and other charge (Q2) of the system and in this manner calculate potential due to all the charges and then add them ,the net potential will the potential due to system of charges

Thus I would assume using a signed distance (1 and −1 in this case) to be the correct approach

This is wrong ! Distance (r) is always taken as positive but sign of charges (Q1,Q2 etc ) is important .

Since the point-charges lie on opposite sides of the origin, I would very much expect them to cancel each other out when calculating the potential at the origin

Why you expecting that potential should cancel each other although I explained above why this is not the case!

previous logic of the sign of the potential signaling repulsion or attraction would not work anymore.

As I explained earlier it still works if you use Superposition theorem but not for system of charges all taken together