Why can't oxygen in furan be sp-hybridized?
Is this only because the overall structure will not be consistent, because sp-hybridization will make the COC line straight, and thus the OCC angles would have to be lesser than the carbons' sp2-hybridization would allow?


Yes, this is the correct answer. A very common misconception is that hybridisation can be used to predict the geometry, or that hybridisation somehow involves an energy cost associated with 'promoting' electrons into the hybrid orbitals. This is entirely wrong.

Hybridisation is always determined by geometry. You can only assign hybridisation states to an atom if you already know its geometry, based on some experimental or theoretical evidence. The geometry of the oxygen in furan is trigonal planar and therefore the hybridisation must be $sp^2$.