Bettas are notoriously difficult to get started on a new food.
It is true that a variety is healthier for your fish.
Ideally you want to feed a protein/staple/veggie diet to any fish, even a carnivore (obviously in this case you want the protein to massively outweigh the others)
to give an example- when I had neon tetras (which I stupidly traded) they were easily the nicest most brightly colored school I had ever seen. This I attribute to their diet:
Flakes,
Live Bloodworms,
Frozen/Freeze dried bloodworms
Spirulina Tabs
Tetramin Bottom feeder tabs
HBH Betta Bites (excellent source of Astaxanthin)
compared to my friend's Neons, which only ate flake with the very rare addition of some freeze dried protein and spirulina, mine blew them away. All water parameters were otherwise identical and even our aquascaping and current levels were similar.