Actually I can explain those pretty immediately and easily; it is still faith shaping the summoning, it's just a different sort of faith.
-Enkidu was summoned accidentally by Gilgamesh in a joke questline, so we're already in a weird loose situation even if Hilidbrand does play hilariously lore-accurately. But Gilgamesh did have faith in Enkidu, as a friend. He genuinely believes that Enkidu is out there somewhere and that they'll reunite someday. Enkidu was also summoned with an actual box of crystals, so we know the power source; it wasn't much, but Enkidu also wasn't much of a primal.
-Ysayle had faith in her own extremely misguided worldview and self-image; Shiva was essentially being summoned as Ysayle's idealized self. You could, in fact, argue that she'd accidentally reverse-engineered the Ancient's Weird Special Combat Forms. This faith isn't quite as strong for her final summoning against the flagship, but at that state she may have also developed a more conventional primal-summoning; her belief in the historical Shiva was always there, and is now taking the stage more than the notion that it's her. They also actually made a point that she only Primals Up into Shiva in areas with naturally high aether resources; usually natural crystal outcroppings. Again this is also a little different for the flagship battle, but there she calls on the Crystal of Light to provide that aether for her.
-Tsukuyomi is indeed an interesting one, but not because it can't have happened by the rules we know; instead, it's more because we don't really know Yotsuyu very well. She might be a woman of faith; it never really came up. But it's also notable that this was a primal who was so inaccurate to the collective image of the figure that it came up in the lore; Tsukuyomi's depicted as very calm, and... well, as a guy, so obviously Yotsuyu was not holding very close to that myth. My general vibe is that Tsukuyomi formed not because of Yotsuyu's faith in the deity per se, but her faith in herself, that happened to form into something Tsukuyomi-esque because of her own associations and the mirror that provided all the aether. I'd compare Tsukuyomi to Shinryu in a sense; not formed out of faith per se so much as stalwart conviction, a faith in yourself and your goals, so as a result the physical form was almost incidental and just informed by the summoner's personal associations. Yotsuyu got Tsukuyomi because of the mirror (and maybe some personal associations with death or the moon), Ilberd got Shinryu because of the dragon's eyes and probably his memory of Bahamut.
Faith can take many forms, and primals actually do a great job of showing that, especially in Heavensward; remember that Alexander was formed not out of faith in a god, but faith that this impossible blueprint could totally work.
Honestly, I don't see what you don't see that says these summons were somehow impossible without some Mystery Energy that, in all honesty, wouldn't have provided anything in these cases anyway.
They didn't. We know because they literally couldn't, their inability to use and manipulate dynamis was a huge plot point, that... for some reason I see people just pretend doesn't exist sometimes.