As a result, it would make a lot of sense if they knew how to perform haki techniques that most characters do not.

Mythical zoans ultimately are only as good as the fighting ability and haki of the user. We see this in both Kaido and Luffy, how their immense haki and fighting styles make these characters as strong as they are. While obviously an awakened mythical zoan would be strong, I think the true power of these characters comes from ancient haki techniques that have been lost in time.

I think summoning is one of these haki powers. In One Piece, the black lightning effect is synonymous with haki, be it armament haki or conqueror’s haki. Hence it doesn’t make much sense to me for this ability to be completely unrelated to haki. Also, the summoning seems quite unrelated to the powers of Saturn’s devil fruit. Of course, he is a mythical zoan and Oda can definitely give him powers that come out of nowhere, but it just seems odd. The eye attack, okay Saturn has magic eyes. He has tentacles that can stretch and poison, as a gyuki would. But summoning? It’s out of place.

I think the way that summoning works is that the users are essentially transforming their bodies into pure haki. That is why the Gorosei seem to materialize from the shapeless black lightning. By turning oneself into energy, you can travel great distances with ease. However I suppose there are limitations to this. You can probably only travel to a place that you can see, as Saturn did when he teleported to Egghead, or to a specific person you have a psychic connection with.

That brings me to the next item.

Psychic connections are definitely a haki power. We saw how Luffy did it to Momonosuke in Wano. The Gorosei have an advanced version of this where they can communicate even when they are large distances apart.

Lastly, there is one more power I think could be haki related. That being regeneration. I think the concept is similar to summoning. If you have the ability to turn your body into pure energy, then you can easily repair your body if it gets damaged. We even see how Saturn has black lightning on his horns when he regenerates.

Throughout Luffy’s battles post timeskip, Oda has often made haki the most troublesome ability Luffy faces. It was Katakuri’s future sight, not his devil fruit, that caused Luffy the most trouble. Similarly, it was Kaido’s advanced conqueror’s haki, not his devil fruit, that caused Luffy to get KO’d so many times.

Given how Saturn’s most troublesome ability is his regeneration, I wouldn’t be surprised if it turned out this was a haki power as well.

With Luffy already having achieved advanced conqueror’s, armament & observation haki, Oda needs to introduce new haki abilities for Luffy to attain in order to beat Teach and Imu. I think it’s possible that this could be part of Luffy’s next round of power-ups.

If I were to guess. I’d say he’d probably get the ability to use divine departure and galaxy impact in order to take on Teach. Those moves are advanced conqueror’s haki only and hence would be useful against someone who can negate devil fruits. But I think for Imu, I think Luffy will get the ability to do summoning, the reverse haki technique, and probably more.