If photons had a very small but non-0 mass, it would simply mean that they actually move slightly slower than c, but it wouldn't otherwise have any significant

If photons had a very small but non-0 mass, it would simply mean that they actually move slightly slower than c, but it wouldn't otherwise have any significant implications. There are in fact experiments that are trying to check whether photons have a mass we can detect. Also, just as a note, I'm using the term "mass" for what used to be called "rest mass" (as opposed to "relativistic mass"). Even if their "rest mass" is truly exactly 0, and their speed is exactly c, they still have energy and momentum, and thus they have mass in general relativity, but that is a different discussion.