my iPad screen will not respond to touch, it will wake, but can not respond to gestures

Follow the all-buttons restart sequence for your iPad here: If your iPhone, iPad, or iPod touch won't turn on or is frozen - Apple Support

iPad models that don't have a Home button: Press and quickly release the Volume Up button. Press and quickly release the Volume Down button. Then press and hold the Power button until the device restarts.

That cleared a stuck screen on an iPad Pro that I recently got to look at... With similar symptoms, too.

If that doesn't work and doesn't clear this and your display screen remains insensitive to touch, then use the Get Support link in the upper right, and discuss your options and alternatives with the folks at Apple.

I had success with the all-buttons reset approach. I did not have success with the usual hold-the-buttons-and-slide method, as the slide would not register. Once rebooted, the issue has not arisen anew.

As for the Apple call queue, yeah, the "best" available approach I've seen suggested was to dig for the chat path.

If you've tried the all-buttons-restart approach and that's failed to clear this, and if you have no case and no screen protector applied and it's still not responding to touch, contact Apple.