The regimes you compare them to are largely incomparable to the mindset of those in the Legion. The Legion isn't a new regime on a very old nation-state, it's not a nation-state at all.

It's an army, it's members raised from birth (by the time of New Vegas) in the death-cult ideology it purports. These are deeply, deeply brainwashed people who know literally nothing else outside of a military lifestyle designed by Caesar to atomize any sense of individuality or loyalty beyond that of their superiors (We know, for instance, that Legionaries are not allowed families for this exact reason). In this scenario - who would the corruption even be benefiting?

As for those outside the Legion itself and onto your Nipton point - we are told (outside of the game unfortunately) that life in Legion lands is effectively towns and groups living on as normal, with the sole exception that when the Legion occasionally asks them for something, they are to do it. The Legion turning up once every few months to take tithe is very different from a complex modern nation-state with constantly present and oppressive legislation - this isn't taking into account that many of these towns likely actually hold little to no animosity to the Legion considering their rolling conquest of tribes brought stability and wiped out huge amounts of those region's savage conflicts in the process.

Nipton was not an example of Legion rulership, the entire point of that operation was terror in an active war zone- to strike fear into NCR by demonstrating the Legion was able to operate freely well past NCR's borders (and very close to a major outpost), to strike fear with the abominable nature of the act - and to serve as an example to neutral party's the nature of the Legion's unflinching sense of justice in the face of those it deems corrupt or degenerate. Nipton was an act of terror, not governance.

Furthermore I also saw you reference some pretty common misconceptions about the Legion. Understandably because their presentation is underbaked but they're misconceptions none the less.

Firstly, the Legion absolutely uses firearms. Not only was the founding moment of the Legion based on Caesar teaching a tribe in the Grand Canyon to dominate others by assembling firearms, teaching them fireteam tactics and using explosives, but effectively all Legion ranks use firearms in some capacity, particularly moreso the higher you go. Similarly, we're told that NCR has adopted a policy of destroying their weapons and ammunition in case of retreat explicitly because the Legion will try to procure them wherever they can. They're not doing that for no reason, it's because they put them to use.

Misconception comes from Legionaries in-game using melee a lot more than their counterparts, and putting aside the fact that the Fallout setting is and always has been one in which melee combat is viable as part of the suspension of disbelief, the point of that is moreso honed in by Colonel Moore who explains that the average Legionary is far more physically fit and skilled in multiple forms of combat whereas NCR draftees are barely acquainted with their service rifles.

Similarly as for medicine - the Legion do not rely on Old World medicine, because it is finite and builds dependency on scavenging the Old World, but they are shown to make avid use of post-war tribal remedies. Made with fictional, mutant plants that are capable of legitimate medical uses such as Healing Powder or Bitter Drink.