Now, the wiper is the one that, in a schematic, is in the middle
Yes, but not all pots have the wiper physically in the middle although most do.
Put the pot somewhere in the middle and measure the resistance from the wiper to each end. It should read half the value of the pot, so for a 1K pot it is 500R from wiper to each end.
If you get no reading then your pot is broken.