To further complicate matters... It's highly likely that a Primal should only be considered "normal" when they aren't summoned. Their default state of existence (where they aren't putting a strain on anything as far as we know) is as some kind of beings in the Aetherial Sea. Only unlike living things that die, go back to the Aetherial Sea as aether and then are reborn as something else, they never naturally leave it. I don't know if it's quite fair to say that primals can't exist without consuming aether as the bulk of their existence happens someplace that is nothing but aether. And the Ascians certainly don't want them staying there as that doesn't further their goals (as far as we know). I think it's more accurate that the primals can only survive in an aether-rich environment and the material realm simply has far, far, far less aether then they need.
As a result, they've got two options, go back to the Aetherial Sea or absorb what aether the material realm has. However, them going back to the Aetherial Sea doesn't help the Ascians (that we've seen). It probably isn't a coincidence that the first summonings we see are of gods that are always supposed to be around in myths. So "leaving" isn't something they are going to want to do. Others of them seem to have goals that they want to accomplish, but it would take too long for them to complete them (or their goals can't be realized for obvious reasons). And then there are some primals that actually do accomplish their goals/purposes. Alexander is supposed to find the best future, it just so happens to be one where he's not around. Susano wants to see which of you are the stronger, which happens. Rahmu needs you to kill him if he judges that you're worthy of the job. I'd be interested to see what would happen if a primal's goal was fulfilled that didn't automatically lead to them being forced out of the material realm and back to the Aetherial Sea.

Headcanon Part

Personally, the most fascinating part of this is how different the primals are in myths vs how they are now. Ifrit was someone who rewarded the strength of the Amaja'a's ancestors, not someone killed of their enemies for example. Titan fills O'Guarmo with ore so the Kobolds can defend themselves. Leviathan brought water to a desert area and died from it. Bismark creates the sky islands instead of destroying them (this difference is mentioned in-game). Given that it was apparently possible to summon primals before Silvertear Skies... there's no reason any of the primals we fight now weren't summoned before. I'm kinda wondering if the "original" Primals in the Aetherial Sea are more like the mythic versions of the primals and are getting more then a bit frustrated because of all the constant summonings. And getting really annoyed that their supposed followers aren't at least giving them the goals/desires they're supposed to have.

Ironically... most of the beasts in the Beast Tribes quests end up doing things that are a lot closer to what the mythic versions of the primals do...