I'm surprised more people aren't complaining about this.
I have the same problem on GT Sport using a G29 with a load cell modification for the brake, but no shifter. Just using the paddle shifters. The bug seems to only happen with certain cars and/or on certain tracks. Sometimes the problem occurs at the rolling race start leaving you coasting as the rest of the field races away, other times it happens only after the car comes to a complete stop. Either way, all on-screen indications of brake pedal, accelerator pedal, and gear selection appear normal. The gear indicator will shift up and down and the engine will rev, but it's as if the transmission is stuck in neutral or the clutch is stuck in.
If the problem is with the G29, it stands to reason that it would manifest itself with all cars and tracks. However, that is not the case. For instance, in the GT League races I can't complete any of the Mercedes AMG Gran Prix races due to this bug. And in the Mission Challenges, the bug happens only on Stage 6-6, the Nissan VGT Ground Combat fighter on Tokyo Expressway. It's worth mentioning that the handheld DualShock controllers still work fine in the aforementioned scenarios.
I've had the bug occur with other random cars after a crash brought me to a complete stop, but haven't kept a detailed list of those. Kind of a pisser if it happens on the last lap of a long race and you have to restart the race. Have also tried running the G29 through a Collective Minds DriveHub, but no change.
If I had to guess, I'd say it's a coding problem with the G29 that for whatever reason only pops up in certain scenarios.