Pardon me, this is basically from memory... and I can't remember how close the T3PA set is to the T500 set in construction.
Couple things...
First, take a look and make sure the pot is fixed securely in place. Be careful not to overtightened it! You'll pull the shell right off the pot.
If you're feeling adventurous while you're in there, you could take a look at the pot and note it's rotation when pressing the pedal. Now, what you want to do is disengage the gear from the pedal such that it and the pot shaft can be rotated. On the T500 pedal set this can be done by removing the plate that stops the pedal and rotating it up and away. Once you get it disengaged, turn the gear/pot in the same direction it rotates when pressing the pedal. ONLY move it far enough that it's advanced one tooth on the gear once it engages again.
If this isn't making sense, don't even try it... I'm not taking responsibility for you breaking your stuff. That said, this isn't hard to do if you're careful.
What is likely going on here is that these pots are probably the same ones that the T500 sets use. If so, only 60 degrees of their rotation actually will cause a change in voltage. Anything before it outputs 0, basically. If for whatever reason the pot was installed with the gear even one tooth off, it could cause something like this. Rotating it a tooth should put it into the "zone." As well, the wheels autocalibrate the pedals as you use them, all the console compatible wheels do. So changing something like this really doesn't hurt anything.
I'm not sure what is going on in this video, it's just an example for you to see what I'm talking about. It's a T3PA pro set and they do look to work the same as the T500 sets.
In the video you can see the gear on accelerator pedal (on the right here) is rotated what looks to be at least one tooth further towards what would be "zero" when compared to the clutch in the video. This is not abnormal actually. But a story for another time...
Anyhow, that's the gear I am referring too. It and the pot shaft need to be rotated such that at rest, it's another tooth further into its travel. It's essentially like having the pedal pressed just a little more all the time. I'd guess if this guy did the same to his set, his accelerator would start working again.