Eventually I would need to explain them, and you'd see they weren't much worse. It's a branch of Linear Algebra. I presume this is operational for a well behaved space over which tangent vectors are definable. Sure, there are infinitely many tangent vectors at a point. This is not so bad.
Linearity keeps life simple.
Now telling me they are local just won't do and nor will telling me they are 'whatever co-ordinates you desire'. There's nothing like a good bludgeoning now and then. So to reward you, I will explain how it works without coordinates. You'll see it's much simpler. As usual, the output depends linearly on each input.