The pronunciation of “route” is largely regional as with other words having differing pronunciations.
OK. Thank you for having made my point.
Even within a region, that word varies with context. I summon my memories of my Western Pennsylvania yute where a white man's form of eubonics known as Pittsburghese is uttered. It's a dialect that follows no known rules aside from consistently being inconsistent. :-)
As for "route", take the case of US Rt. 19. variously known as Perry Highway, McKnight Road or Warshington [sic] Road. It was universally called "Raut Nineteen" by everyone I knew, friends, foes, family, colleagues and television announcers. In contrast, Saw Mill Run Boulevard or PA Rt. 51 was referred to as "Root Fifty-One".
I've called upon a couple of 'Burghers (and those with some ties, even tenuous, to the area) to confirm or deny the correctness of my recollections since I've been gone from the tri-state for over four decades. My memory may be slippy 'n'at. Yoi!