I think this is the good/bad thing about the political situation in the Sequels.
There are multiple ways we could define and interpret the political situation in general and more specifically within the opening crawls, but according to what we know in the canon, there’s some truth in most of these interpretations.
Let me lay out some of the interpretations:
The New Republic supported the Resistance, but it didn’t support it at the same time.
Seems contradictory, but the truth is that some senators within the Republic (Populists) did support Leia’s Resistance movement, while others (Centrists) did not. So a crawl could say “with the support of the Republic”, or “failing to convince the Republic” and both would be true to a certain extent. Even though Leia didn’t have the Republic’s full support, some Senators did.
I don’t know the source, but I also feel like I read somewhere that some Centrists worlds actually left the Republic and aligned themselves with the First Order a few months before TFA, which seems like a pretty big deal but nothing in the films suggests this, so an editor could either run with that idea or completely ignore it.
That’s what’s funny to me, the theatrical crawl says the Republic supported Leia’s Resistance (openly? secretly?), but in a deleted scene Leia also says, “Not all the senators think I’m insane… or maybe they do… I don’t care!”
This seems to imply the opposite, which lends more to the idea that the Resistance might be more of a rogue movement that doesn’t have the Republic’s official support.
You also wonder what the Resistance technically is. Are they an anti-fascist spy network that is spread across the galaxy, or are they more of an organized militia within First Order space? The latter would make them seem like the French Resistance within Nazi-occupied France during WW2, but the theatrical film seems to imply the former more, so the “Resistance” refers more to their goal of wanting to resist the growing influence of the First Order. But honestly you could probably go with either option and what exists within the film wouldn’t necessarily contradict either interpretation.
As for why Leia formed the Resistance rather than just operating within the New Republic military, you have a few options which all seem to be partially true:
The Republic did not want to start another war (tired of war/preferring diplomacy), and/or they didn’t have enough proof that they were more of a threat than other past Imperial remnants, so Leia took matters into her own hands to prove to the Republic that they needed to take action before it was too late.
The Republic did see a threat, but investigating the First Order for war crimes in an official capacity could be risky (maybe lacking evidence), so the Resistance was meant to be a covert militia to gather enough proof that the Republic to act on (at least some pro-Leia senators had this mindset).
Leia knew there were First Order spies within the Senate feeding information to the FO as well as downplaying their threat to the Senate, so Leia formed an independent covert militia to avoid being spied on, as well as finding another evidence that the First Order was preparing for war against the Republic.
Regarding the First Order, there are multiple ways you could elaborate on them as well. The theatrical crawl just says they have “risen from the ashes of the Empire”, which is pretty vague. Some phrases I have suggested in the past have been that they have “emerged from uncharted space”, more recently “[they] have taken over the remnants of the Empire”.
I think the biggest question regarding the First Order is how did they consolidate power and resources under the nose of the New Republic, and I guess a good crawl would at least hint towards that. Saying that they have emerged from hiding, or have absorbed other Imperial remnants could help explain that. The funny thing is, either interpretation seems to be partially true in canon.
The First Order:
-has been hiding in the Unknown Regions rebuilding their military in secret.
-has absorbed the manpower/resources of most of the remaining Imperial remnants.
-has wealthy benefactors within the Republic supporting them (like pro-Imperials or war-profiteers).
-has spies and sympathizers within the Senate helping downplay their scale and overall threat to the New Republic.
-were being led by a mysterious Force-user powerful in the dark side that helped them navigate the Unknown Regions and used its secrets to help them rebuild.
All of these factor into the First Order’s backstory in canon, as far as I know.
While all of these interpretations of the political situation between the New Republic, the Resistance and the First Order does bring up a lot of interesting ideas and nuance, because none of these things are really defined well in the films, the whole situation seems vague and confusing. While it is good for us that we can find our own interpretations, I think it is a weakness of these films that they don’t just pick one of these interpretations and go with it.
So what we’re stuck with is trying to decide on which direction to go in (and these are only the canon-friendly interpretations, an editor could pick even more). Obviously each editor is going to have their own preference, but it would be nice to determine which interpretations fit best within the story and themes of the Saga and Sequel Trilogy. I guess they all have pros and cons, but maybe it is a matter of finding one that has more pros than cons, if that makes sense.