VFR towers (which won't tell you "radar contact") can relay radar
vectors from the radar approach control which serves them.

I can buy the idea of a relayed vector, but the person who provided
the vector still must see you on radar. When the aircraft is sitting
on the ground, approach control can't see it and therefore can't
vector it.


Consider that if you take off with the heading and start to wander
into some obstacle, tower won't be able to do anything about it
(because they won't know) and neither will the approach control until
they get you on radar, which will happen at various altitudes,
depending on where you're departing from.