I think there is a widely held misconception about the cause of induced drag. The wingtip vortex itself is not the cause of induced drag, per se. The cause of induced drag is the distribution of downwash along the span of the wing. Downwash is the slight downward component of velocity of the air approaching the wing, caused by the very act of generating lift. You simply cannot prevent induced drag by making it more difficult for air to spill from the bottom of the wing to the top. To say that a square tip has lower drag for this reason is to misunderstand the phenomenon. It is possible that some wingtip designs could reduce induced drag by shifting the distribution of downwash ( usually this means that the tip design pushes the center of the vortex out from the end of the wing, emulating a longer wing ).
Induced drag is caused by spanwise flow, inboard on the top of the wing, and outboard on the bottom. This spanwise flow is caused by the pressure differential between the top and bottom of the wing. So, you could say that induced drag is a byproduct of having to use finite wings... as an infinite wing has no induced drag.
Induced drag, especially at low speeds and/or high angles of attack is definitely the largest component of drag on an aircraft (neglecting difficult-to-quantify drags like gear, flaps and other protuberances or high lift devices).