The solution is laughably simple, but you wouldn't necessarily think about this. Just head into the graphics settings of the game and make sure that the resolution of the game is running at 60hz. That's literally it. I, like many people today, play at 144hz. Turns out this causes issues for the achievements script or whatever in GTA IV.
If you wanna make double sure that everything works limit the FPS of the game to 60 (via Nvidia Control Panel or any other similar tool) and enable V-Sync in the graphics settings.
After driving around for hours at the airport and losing like 30k to hospital bills for dying at the Swingset of Death I managed to get the achievement in under 2 minutes at those yellow ramps at the airport, just because the game doesn't work as intended at 144hz.

To elaborate further, this is really about the refresh rate, not just FPS. I regularly capped the framerate of GTA IV at 60 FPS because the stuttering in this PC port is so bad. That alone does not suffice, because I already thought about the framerate causing issues for the achievements and tried to trigger the achievement after capping the game at 60 FPS. You need to set the game to 60hz OR IT WILL NOT WORK!