What are Omnisource Crystals used for?

That makes them a different class of material from everything else you spend on stats. Stardust covers capability awakening, potential upgrades and IV stat modification, but all of that works inside the ceiling the game already hands you. Omnisource Crystals are what move the ceiling itself. So the moment an Aniimo you have invested in hits its IV maximum, this is the material that matters and nothing else substitutes.

Worth keeping in proportion: resonance is one track among several. An Aniimo gets stronger through standard leveling, traits, potential spent as Capability Points, resonance progression, carried equipment and runes. Omnisource Crystals sit inside that one track, and a team that is under-leveled or running no runes will not be rescued by them. They are a late refinement on a creature you have already committed to, not a shortcut past the rest.

Not the other crystals

Several materials in this corner of the game carry "Crystal" in the name and none of them are interchangeable. Resonance Star Up runs on dewdrops or Dewdrop Crystals together with an Elemental Stone or evolution crystals, so a stockpile of dewdrops does nothing for the stat upgrade side of resonance. Lumin Crystals are something else again: the premium currency you buy with real money to spend on cash shop items, a wallet rather than an upgrade material.

When you are reading a cost screen, check which crystal it actually wants before you spend.