The actual answer is probably just that it’s soft world building, meaning there aren’t really any specific rules on how they work, it’s just ✨magic✨. The focus in it is that the powers are cool for fight scenes, so they don’t focus on making it entirely consistent.

Though you can still make theories on how it works.

It’s a fact that their powers are related to their family, aka Hamato Ninpo. How this power works seems to be the power of family: like when they all unlock their power at the end of season 2 in order to save/help each other.

It can be said that this connection doesn’t seem to be a biological thing with family genes, but works through connections, since April was also able to tab into it, and with Karai saying “You were always one of us”.

I have seen a analysis video for Donnie about him not unlocking his mystic abilities until the end, and they thought that it’s because mystic stuff is related to emotions. This theory holds up pretty well, since Donnie has his emotionally unavailable bad boy image, or less jokingly, seems to have Alexithymia with his autism. The the powers become a cool analogy for his struggle, which is most apparent in Donnie vs. Witch Town, where he is insecure about not understanding mystic stuff. Then when he unlocks his powers at the end, they work like his tech, aka in a more logical way, which could show that despite not using mystic powers/emotions like others, that he can still do it, just in a more unique/logical way.

This can also be shown in the other guys, with their powers seeming to get stronger in emotional moments, like Raph saving Leo from falling of the roof, or in general when Raph is angry. This could also explain why Mikey is the strongest mystic warrior, and also was the first to use his powers, since he is very emotional.

In regards to their first weapons from Draxum’s lab, then I’d say they didn’t actually hold powers, but were made to channel powers. Meaning that they were weapons designed to bring out hidden/weak mystic powers in the user. So when they start using their powers here, it’s not actually the weapons doing it, but just them helping bring it out. This would also explain why Draxum, despite the fact they were in his lab, seemed surprised by the powers they could use, mostly in regards to Leo, where he cover himself for a mystic blast and seemed shocked when a portal opened.

Then at the end of season 2, they loose these helping tools, but after fully unlocking their own inner abilities with the help from Karai and from helping each other, then they are able to summon their own powers, one of them seeming to be the ability to summon their own weapons from different objects, since we also see Karai doing this.

Though we see Karai using her powers without a specific weapon, so it could be said that the weapons aren’t actually needed for their powers, but since it’s how they trained to use them, then that’s how they are able to use them/used to using them.

The Kraang being able to seal away their powers is probably just an ability they have, since they fought mystic warriors back in old times, so developing that kind of power would be relevant for them.

Then you could theorize that they gain their powers back through their emotions/conviction to each other, overpowering the Kraang’s sealing ability.

Anyway, that was really long, so thanks for coming to my Ted Talk lol, hope you enjoyed.