Without looking at any sort of diagram, I'm going to say no.
If any thing, an unresponsive, disabled, or disconnected oil pressure sensor (or gauge) should do nothing more than trigger a warning light on the dash.
Of greater concern would be the possibility that the oil pressure is actually low, and the gauge is reading correctly.
As far as the no-crank-no-start (NCNS) condition goes, there are many possibilities. Primarily, I'd be checking the battery terminals and ground wires. If those check out, make sure that the starter's wiring and harness are connected securely and grounded (a multimeter might be handy for this), and if all the wiring checks out and you still get NCNS, it might be time for a new starter.
There are other conditions that can cause NCNS, including a poorly-seated starter fuse (you know it isn't blown), worn out or damaged relay (present doesn't mean functional), break in the wiring behind the fuse or relay boxes, or some thing simpler such as the shifter not fully seating in Park or Neutral. Some times the obvious things are the tricky ones.
Crank no start could be caused by a number of things; worn/bad fuel pump, clogged injectors, bad or poorly connected spark plugs/wires/ignition coils, or simply a worn-out starter that spins but doesn't grab. NCNS is usually either electrical disconnection or safety feature. You mentioned CNS before NCNS, so it could be some of each, or some thing else.