I have the same problem with the Jaguar, it has that gold line on the car in the selection menu but I don't have those two mods even though I did the race multiple times.
My issue with the Lancia is that it doesn't have the gold line, even though I've got gold in all the events.
I've had the issue with the missing short gears and aero for the Jaguar as well, but I've found a solution for that one. The problem is that after you've installed the Multiplayer Update 2 or the according DLC, gold in the Burning Rubber event doesn't give you short gears and aero anymore, it gives you the ram chassis, so there are now two events which give you the ram chassis and none give you short gears or aero.
What you need to do if you desperately want this fixed, is temporarily uninstall the Multiplayer Update 2 or the according DLC, then decline the update and play without it. You'll get short gears and aero back.
Now, before you re-install the update, you need to make sure you unlock both pro mods, otherwise, they'll be gone again after you re-install the update. The pro mods, however, remain, as these aren't tied to the Burning Rubber event.
And yes, it's quite a bit of effort if you need to uninstall and re-install the according game data, though there's a shortcut if your game save and your game data are in different locations, as you can just temporarily unplug the game data, if applicable, instead of removing and re-installing it. Worked for me on my XBox with game save on internal memory and game data on external drive.
Edit: And no, I didn't see any issues with my game save after re-installing the update, all progress was still there, even the stuff you need the update for.