It doesn't make sense that a diverse base of Purge Supporters doing it for various reasons (racism, anti-immigration, anti-class, etc) would be organized and well-equipped enough to overthrow the military and hence the government shortly after. In real life (Jan 6, 2021) an uprising was still squashed pretty quickly despite the terrible security and no more major uprisings occurred after that. The movie didn't explain how the Ever After Purgers succeeded, but it would make sense if: 1) At least half of the military themselves are siding with these rioters, which the media had covered up to avoid showing the weaknesses of the New Founding Fathers' position. Or 2) alternatively, the movie could've neglected to explain that the country's resources had been exhausted from fighting another war outside of America, leaving the military ill-equipped to handle the uprising... Or 3) The Ever After Purgers are political tools organized by Anti-NFFA groups and mass-coordinated to overwhelm the country, although this scenario might not make sense since it would mean that National Security are headed by complete idiots for not squashing it before it began.
There are about 2 million members of the United States Armed Forces, including National Guard and Reserves, with about 1 million being on active duty in the United States and very few of these would be front line troops. If even 1% of the population are Ever After Purgers (and as the NFFA has spent decades flooding the airwaves and the internet with pro-Purge propaganda, it is likely much higher), then the government is going to be seriously outnumbered.
To add to this, there's a non-negligible chance at least some people in the armed forces sympathized with the Ever After Purgers, and helped either delay, sabotage, or infiltrate army / National Guard / Police, making the initial battles swing harder in their favor.
The end of the movie is also about 36 hours after the legal end of the Purge, so it is not surprising that the government is still getting its feet under it a deal with a nation-wide insurrection.
How long would this forever purge actually last? The population of the US relies on transport and other complex infrastructure for food, water and gas. Even if they slaughtered 90% of the population, within a few days, the Ever After Purges would be getting around on foot and horseback and struggling to find food.
This is pretty much Truth in Television for similar ideologies, though: they can't logistically work, and ironically the only reason they can keep existing is that they never reach power and are forced to deal with their failures.
Besides ending early on in the movie, the actual Purge seems overall pretty tame for the series' standards, with no main character even threatened by it, despite the fact that all of them turn out to be targets for the Ever Afters. It's very likely, however, that the Ever Afters either did not take part in the Purge or did it in a low-key way, keeping their targets to those who were easy pickings on the streets rather than hunting them down, using the time to organize and prepare. After all, why take the chance when they are expecting it and protected, if they can just wait the next morning, when they will be defenseless and unprepared? When Kirk takes the Tuckers hostage he specifically references how they are not protected by their steel walls anymore.
The latter hypothesis seems to be the case, with emergency services caught completely unprepared and the deputies who arrested Darius and Adela thinking it's "just" people who haven't heard the end-of-purge siren. In as much as this seems like a country-wide conspiracy and plan, waiting for the "undesirables" to leave their fortresses seems a logical move. In fact "we can't purge like we want because the people we hate know we're coming and protect themselves" would be a prime reason the Ever Afters would want an endless purge.
How were the NFFA able to be reinstated in power? Well, the previous movies established that under the NFFA the usa weren't actually doing all that well, with the Purge serving (among other things) both as a distraction and as a way to direct the blame elsewhere. Roan might be a good politician, but that's not enough to undo the harm done, particularly if faced by the NFFA and the hostile infrastructure left behind obstructing her. The NFFA can just go back to their fear-mongering, promising the people someone else to direct their blame and easy solutions rather than having them face their problems.
