Hope you don’t mind a long response 😊
I’ve always seen Starfleet as pro their members having religion if they wanted to but that it shouldn’t be allowed to interfere with their duties as members of the fleet. I suppose I think of Starfleet as being modelled on the original civil rights movement in America: everybody working together based on merit and judging people for the content of their character and not the colour of their skin (or their planetary origin).
Picard’s crew practice elements of their religions, for example Ro Laren is given permission to wear her Bajoran earring, Worf gets time to go to Borath to spend time with the monks. What Picard doesn’t do is let crew’s cultural or religious beliefs interfere with his mission - Picard even tells Worf at one stage, after Worf kills Duras for murdering K’Ehleyr, that there are representatives of X number of cultures on the Enterprise and he expects them not to let their cultural beliefs interfere with their duties as officers.
I always see Picard himself as interested in other species’ religious and cultural beliefs, often as part of his (intermittent) interest in archaeology. He respects other cultures even if he disagrees with their practices - he doesn’t try to stop Lwaxana Troi’s boyfriend from euthanising himself in line with the values of his culture, he doesn’t try to stop Riker’s girlfriend from undergoing conversion therapy, he is willing to let Worf commit ritual suicide after he sustains a paralysing spinal injury.
Picard sees himself as unfit to stand in judgement of others. In a lot of cases like these, Kirk would have beamed down to a planet, pointed out where they were wrong, and stopped whatever cultural practice they were doing that he felt was bad, particularly if it was a threat to his crew - he believed in universal moral truths. Picard was more into relativism.
When it comes to DS9, I always viewed the Ferengi stories as a critique of people who believed American free market capitalism could solve everything. They were a great contrast with the Federation’s post-scarcity societies. Although, Quark had an astute view of humans: “They're a wonderful, friendly people as long as their bellies are full and their holosuites are working, but take away their creature comforts, deprive them of food, sleep, sonic showers, put their lives in jeopardy over an extended period of time and those same, friendly, intelligent, wonderful people will become as nasty and as violent as the most bloodthirsty Klingon.”
And, if I’m honest, I find the Bajorans annoyingly religious - being Irish I understand how culture and religion play an important unifying role in societies who are fighting against colonisation, but I still find the Bajorans OTT. I also think Sisko really enjoys being the Emissary a little too much and begins to think too highly of himself and his own decisions, especially during the Dominion War.
Having said all of this, I don’t see NuTrek as being able to tease out the cultural and moral nuances about the various issues raised in older series of Trek. And the one thing I loved about every series of Trek was the examination of morality and viewing elements of our own culture through the lens of an alien race’s behaviour. And I liked the fact that often there was no one right answer presented to the viewer.