The light is redshifted when it "climbs out" of a strong gravity field but still moves through space at a constant speed. https://en.m.wikipedia.org/wiki/Gravitational_redshift
2nd stupid question: I though gravity acting on light was a 'light as particle' thing, rather than 'light as wave' thing? If that were the case gravity acting on light as particle manifesting in light as wave doesn't seem consistent.
I think the best approach is to be pragmatic and simply follow the observations and what the experiments tell us rather then force it into one paradigm or the other and complain when it doesn't make sense. This is sometimes referred to as the "Shut up and do the math" approach.
In some ballistics problems, you'll consider a flat earth (baseball problems) in some others a spherical one (spaceflight). It is about which model makes the math simpler.