I had the exact same problem, air would stop blowing from center vent when I pressed on the accelerator. How I know how to fix it, Happy Again
Remove the glove box, 2 screws at the top edge next to the latch. 2 on the bottom, sometimes they are cover with a plug, they are right where the felt part of the glove box starts and about 1 1/2" from the left and right edge of the box.
Pull down on the top of the box and outward, it will pop out,
!!! UNPLUG THE GLOVE BOX LIGHT BEFORE YOU PULL IT TO FAR !!!
You should be able to see the top of the Switchover Valve Block (SVB), at the top there is a Green line. Right in front of it is a black tube connecting the right vent and the center vent air (so air can flow to the right side vent). Push it to the left hard and remove it from the right vent connector. Now you should be able to see the whole SVB. On the #2 hose (Marron or Dark Red- solid color) you want to clamp that line off. I used a bent washer, fold the washer, place the line in the fold and clamp it shut with a pair of Vise-Grips. I used two, just to make sure. NOW THE AIR CONTINUES TO BLOW OUT THE CENTER VENT.
This may effect your defroster, I used the switch to change air flow and I am getting some air to the Defrost vent when I have it in the ARROW UP position.
Use the AUTO air and you will have air flow regardless of pressing on the accelerator.
You you would like to test this concept first, just put a pair of Vise-Grips on the Maroon (clamp it closed) and give it a test drive. Then you can work on clamping it off with the washer method I used or whatever you can think of.