Obtaining a finite-time blow-up for Navier-Stokes does not necessarily advance the field of mathematics by any significant measure, whether the proof is very long or very short.

As a concrete example, such a proof could be less than a page with very specific initial and boundary conditions and inserting them into the equations to get something that goes to infinity when time goes to some finite value.

This would resolve the Millenium problem but not make humanity any smarter.