Can't get my possessed car to move...

So the Vehicles template on 5.2.1 is fully setup input wise to drive as soon as you launch it and press play in editor. So something else must be causing your inputs not to land.

So the input context I showed off is (supposed to be) already in the default player controller, in your case it looks like you put it inside the vehicle itself, it shouldn’t require this due to being a child of the player controller when in game, so you can revert the vehicle BP to the way it was before!

Though your answer here gives me a solid hint of where your issue may be:

It prints the possession report twice, in fact.

Did you by chance place a vehicle into the map? As the game mode (as it begins normally) spawns it’s own at the player start location. So you may in fact have one you placed and one that’s spawned.

For example:
image

This spawns one car and the other just sits in the world. It takes possession first, then the spawned one takes possession one tick later.