As for baggage I'd love for them to get rid of:
- "No money."
Absolutely, and it would be really nice if fans would stop acting like it's the literal truth and that everything is free. It really isn't. No civilization the size of the Federation could survive without some kind of economy, which means
something is used as a medium of exchange as legal tender on all Federation worlds, space stations, and starbases. Even Voyager had an internal ship's economy, since credits would have been useless so far from home. As Kirk said, it's not
cash, but that's nowhere near saying the society itself doesn't use money.
And if Picard thinks so, somebody needs to remind him that Beverly had to pay for that ugly piece of cloth on Farpoint.
Maybe we don't, but unfortunately, the shows did, for the same reason that TOS needed improbable Earth-duplicate gangster planets and Nazi planets and Roman planets: to save money by reusing existing props, sets, and costumes from period productions. In an ideal world, we wouldn't have had any of these things, but commercial television is a realm of compromises, not ideals.
One could try to discard that baggage in a new production, but we'd probably end up with new baggage, like a galaxy where every planet looks like the woods around Vancouver.
Is there something wrong with the woods around Vancouver?

For me, it's nice to see places I recognize from having been there in person. And at least it's more imaginative than the same gravel pit used over and over in Doctor Who or the same exterior apartment building used for both Rose and Clara.
For me:
Religion: The idea that humans literally become secular over night isn't realistic. Religion is as old as humanity itself, yet somehow, it's shed by the 22nd or 23rd century, and the only religious people left are aliens? There should be Earth religions that are still prominent. I think it would be interesting to see how space travel and aliens affected Christianity, Islam, Judaism, and indigenous cultures across Earth.
Hardly overnight. And I can think of several TOS characters (regular plus guests) who gave the impression that they believe in God/Jesus. Just because Picard makes a speech about how humanity doesn't do or believe this or that, that doesn't make it true - it's pretty obvious how much of an insulated life he leads, away from the normal concerns of people who are not part of Starfleet.
Time Travel: An over used trope. Simply put, it should be nearly impossible to achieve, but by Voyager, it is established it becomes so common, they actually have a temporal police?
I never saw most of
Enterprise, so I'm completely unfamiliar with the Temporal Cold War. But it seems reasonable to assume that except for the few instances we saw in TOS and Voyager, organized,
formalized time travel became commonplace centuries in the future and worked its way backward. It's not like time travel became an institutionalized thing in the 23rd or 24th centuries. At that time, if anyone time traveled, it was either by accident or due to extraordinary circumstances - hardly an easy thing to do.
Transporters I agree make escapes a lot less interesting but they are a big part of Trek aesthetic. I say keep the transporters but require a transporter fixture on both ends. No site to site transporting.
So to transport anywhere, there would need to be at least a single-person transporter pad already established at the other end? How is that to be set up beforehand? If the personnel and components have to be shuttled down/over/whatever in order to set up a transporter, what's the point?
Over all, there was something of an anti religious attitude in Trek, mainly TNG. That statement by Picard about religion inevitably leading to superstition and dark ages is one example.
But that was Picard specfically referring to the Mini-Taken's past religion and not (iirc) religion in general.
"Mini-Taken"?

Do you perhaps mean the Mintakans? That's one of the few times TNG used the name of a real star, btw (Mintaka).
Kieko O'Brien didn't want to compromise at all about Bajoran religious beliefs and the class room.
I think that was more Kieko arrogantly imposing her perceived cultural superiority upon the Bajorian children, She didn't want to employ local terms in her lesson plan, even when (initally) politely asked to do so.
It's not arrogant to use proper scientific terminology in a science class. After all, if you were teaching a group of RL elementary students about thunder and lightning, you wouldn't tell them that lightning was caused by Zeus throwing a temper tantrum and thunder the result of the angels holding a bowling tournament, would you (seriously, somebody said that to me once about thunder

)? At least I sincerely hope that's not what the kids would be taught.
^ But the indigenous terms were heavy with religious meaning
When Kai Winn entered the room, Keiko addressed her in front of the children using her religious title of Kai. Kai would be an example of a indigenous term.
"Kai" was a Bajoran honorific, much like "Sister" is for a nun. And there's a great deal of difference between using an honorific and muddling up a science lesson with stuff that gets in the way of learning about science.