I would like to know if painting a tile roof is a must or is it just cosmetic? Should I just repoint the roof and the ridgecaps?
Simply using paint is a cosmetic application. Using a silicone or other sealant based paint would actually improve water protection. If the goal is to seal the roof then it might be a good idea to apply silicone sealant and then paint. For cosmetic applications, any outdoor latex or acrylic would do just fine.
It depends what you're after, do you just want to be dry & are fine with being the rat house on the block...unless you're the only house on the block...but what would Mama say.
Seal it & then Paint it! And yes, that includes your masterful repointing...you want that to last too, don't you. You could even go with a Thermal or Ceramic Paint that will last much longer & actually give you something close to a radiant barrier that the roof is perfect for, but totally not worth doing.
Paint here might just possibly look good one day one. It has zero chance of looking good after the first winter, summer or passage of any entire season of any reasonably popular TV show.
If you want a roof that will last, replace those shingles with lasting roofing material. Asphalt shingles are the most common, but don't last nearly as long as the alternatives.