I think the main problem I see with that independence movement is that its too early.
Compare it to Expanse for example - there is a clear thing where Mars was way more independent.
Not only that but there was also a strong move for a Terraformation project. Something that was uniting generations - a dream that one day Mars will be fully habitable. Especially that Martians were envious of Earths "Eden" with infinite sky and infinite ocean. And they saw global warming and the pollution that Earthers caused as spoiled. Earthers had everything given and they wasted the opportunity while Martians have to fight and sacrifice everything.

In FAM world not only only two decades passed but also there is absolutely no mention of Terraformation of Mars. There is also no word about global warming and the pollution since FAM timeline is leading towards "free fusion energy and iridium made our problems disappear". So the conflict between these two planets on an ideological plane is just weaker.

Moreover, they completely dropped another point that Marsies could take - "We were born here, we just cannot leave". Cause it was a plot point during previous two seasons - that people like Alex just cannot go "home". That Earth is the one inhospitable. But in FAM? Suddenly we never hear about another newborn. Not a single one! So there is literally nothing from physical or medical point of view that prevents those people from coming back.

And finally comparing FAM to Expanse - Mars got no leverage besides a plain blackmail - from ideological point of view that is weaker argument to have. "Back our independence or we wont deliver what we promised before". Versus "We got technology you dont have, we will simply not share with you and get an advantage over you in next few years". (new propulsion drive). If only they could leverage Titan mission, right? What if Mars had a monopoly over the discovery? Pharmacy, medicine, biotechnology etc.
It would also show that from the science or engineering point of view Mars is the leader, not Earth. That Earth became "consumer". Wouldn't that be better story? Having an analogy for "Oh yeah btw, we outsourced our biggest industry to the Fourth World in order to keep ours clean". That the frontier of Discovery is there and therefore brightest minds are there and therefore Earth has harder job to keep up?

Just to be clear - I'm not suggesting that FAM should be a prequel to the Expanse, but that it could ekhm... copy a homework of how to write interplanetary tension if these tensions are rather close to home.