I have to say that I think the current popular idea of "we know how X drug works in the body because it does Y to Z" is patently insane. It may be useful information, sure, but it's not really _why_ it works, because we don't understand how the body works. It would be like Toyota saying: "the Corolla works on gasoline because it burns inside the engine, but we have no idea why the engine breaks when you put acetone in it." It would be obvious that Toyota had no goddamn clue what was going on in the engine! And yet, that's exactly where we are with most drugs including lithium; most of these so-called mechanisms would work on rubidium, so why does rubidium have different effects? Who knows!
If you read a literature review for lithium's mechanism of action, it's a wild ride [0] that clearly demonstrates exactly how little we know. More importantly, though, is that all of this is made up ex post facto. Nobody can take a _new_ drug and tell you anything concrete about what it will do or how it will work: instead, we look at what happens and _then_ we make crude guesses. It is essentially modern miasma theory. For those who have forgotten, the application of miasma theory built our first sanitation systems, which eliminated more deaths from disease than the entirety of vaccines. That is to say: just because we are groping in the dark does not make the work useless, and indeed, our first vaccines were also constructed with very primitive methods.
I think that a more concrete understanding of the human body can only come when we start to understand all the many pathways of life in/on/around the body. Right now, our technology constrains us to investigate only individual points in mostly static ways. Give it a few decades of advancement and I bet we'll have some fascinating insights. I would also bet that, as with any complex system, there will be no simple answers to how things work.
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For what it's worth, in the specific case of lithium, the American Society of Health-System Pharmacists would agree with me that the mechanism of action is unknown: https://www.drugs.com/monograph/lithium.html
[0] https://link.springer.com/article/10.1007/s40263-013-0039-0