There are larger minds than ours, and they've been well-attested for millennia as celestial entities, i.e. spirits. Approaching it purely within the realm of the kinds of minds that we can observe empirically is self-limiting.

Well attested to is different from convincingly attested to. You may have noticed that people will say just about anything for lots of reasons other than averring a literal truth, and this was particularly true for those many millennia during which human beings didn't even know why the sky was blue and almost literally could not have conceived of the terms in which we can presently formulate an explanation.

The "people in times before the enlightenment just attributed everything to spirits because they didn't understand things" argument is tired and boring. Just because you're not convinced doesn't mean that it's not true, modern man.

That isn't my assertion. I actually think people in the past probably did not seriously subscribe to so-called supernatural explanations most of the time in their daily lives. Why I am saying is that its quite reasonable to take a bunch of incoherent, often contradictory and vague, accounts of spiritual experiences as not having much epistemological weight.

Then we disagree about the basic assumption. I do think that people throughout history have attributed many different things to the influence of spiritual entities. I’m just saying that It just was not a catch-all for unexplained circumstances. They may seem contradictory and vague to someone who denies the existence of spirits, but if you have the proper understanding of spirits as vast cosmic entities with minds far outside ours, that aren’t bound to the same physical and temporal rules as us, then people’s experiences make a lot of sense.

Okay, you have proposed a theory about a phenomenon that has some causal influence on the world -- ie, that there are spirits which can communicate with people and presumably alter their behavior in some way.

How do you propose to experimentally verify and measure such spirits? How can we distinguish between a world in which they exist as you imagine them and a world in which they don't? How can we distinguish between a world in which they exist as you imagine them and a world in which a completely _different set of spirits following different rules, also exists. What about Djinn? Santa Claus? Demons? Fairies?

We can experimentally verify spirits by communicating with them. Many such cases.

Now, do you mean measure them using our physical devices that we currently have? No, we can't do that. They are "minds beyond ours" as OP suggests, just not in the way that OP assumes.

Djinn: Demons. Santa Claus: Saint (i.e. soul of a righteous human). Demons: Demons. Fairies (real, not fairy-tale): Demons. Most spirits that you're going to run across as presenting themselves involuntarily to people are demons because demons are the ones who cause mischief. Angels don't draw attention to themselves.

I don't know, it seems reasonable to conclude that the experiments you describe point strongly to an endogenous rather than exogenous source of these experiences, especially since people who have these kinds of experiences do not all agree on what they are or mean and the experiences are significantly influenced by cultural norms.

An electron is a bit like a demon in the sense that you can't see one directly and we only have indirect evidence that they exist. But anyone from any culture can do an oil drop experiment and get values which aren't culture bound, at least in the long run. People have been having mystical experiences forever and the world religions still have no agreement about what they mean.

Seems implausible to me.