There is a lot of discussion about what wizards are capable of in a fight against the muggle world, but many of those capabilities rely on things which wizards either didn't use, or didn't use much of. Could a wizard infiltrate a military base while invisible, inaudible, immune to toxins and physical force, and carrying an AOE weapon capable of killing every soldier there? Yes. There is no reason why a wizard shouldn't be able to do that. Thing is, it requires the use of a large number of expensive and rare resources (potion ingredients, demiguise hair, etc), difficult or little known spells, and training with both the AOE weapon, and infiltration techniques. Could wizards overcome those challenges? Yes. In the numbers to matter? Probably not.
There is a reason that every auror and death eater doesn't wear enchanted armor with an invisibility cloak over it. Its not that those things are superfluous, its that they are expensive, difficult to make, and don't provide enough of an advantage in combat when fighting other wizards to be worth the expense. Those factors don't change when going up against muggles, even if the utility of the equipment goes up. Just because invisibility cloaks are more useful against muggles than wizards doesn't mean that they will suddenly become cheaper, easier to make, or become generally more available.
We know most Ministry wizards had a horrifically bad DADA education at Hogwarts and most were unable to cast a basic shield charm (the Weasley twins mention this explicitly when talking about selling shield hats). Could all these wizards who are effectively untrained civilians be taught basic combat spells and maybe even useful spells to create enchanted armor, shield hats, etc? Probably. But we know in cannon that the Ministry was not putting their bureaucrats through DADA refresher courses even while dark wizards were terrorizing the country, so presumably such mobilization and training was deemed impractical. So in a hypothetical war situation, the ministries will be relying on the same people they have always relied on, a handful of well trained law enforcement personnel, and whatever auxiliary volunteers they can muster up in a crisis. Bluntly, the wizarding world does not have crack troops. They don't even have soldiers. They have police officers and gang members/terrorists as their only effective fighting forces. They do have a handful of heavy hitters, but you can only deploy your Dumbledore's and Moody's and Voldemort's so many places at any given time, and they can only do so much.
The wizards if fully mobilized, united, and trained for war, with ample preparation and time to build up, would utterly crush any muggle army, country, or even planet. But that is not what we are talking about here. We are looking at a society which is loath to stir itself to war, divided along numerous social and political fault lines, utterly untrained for war, without any form of military institutional knowledge nessesary to get trained, and with limited preparation for a war with muggles. The only thing the wizards have had, is plenty of time to build up, but since they have apparently not done so, such time has been wasted.
Wizards are fully capable of subverting a government, even as is, but actually fighting an open war? They would be so absurdly outnumbered that they would actually need to send lone wizards to take control of entire military bases. And every one of those missions would need to go perfectly. And if even a handful of wizards got wind of the plan, and either warned the muggles or worse, give them aid, it would never work.
Wizards are a lot like aircraft in terms of a modern war. They are highly mobile, incredibly destructive, and can turn the tide of battle in a sortie. But what they can't do is occupy. They can destroy the enemy, but they can't win the war. Unless they are literally fighting a war of extermination, the wizards will not be able to control the countries that they subvert or conquer.
Lets say the wizards take control of the government. Okay, lets say that the top 200-300 government officials are under wizard control. Great. How do you control the hundreds of thousands of bureaucrats, the tends of thousands of corporations, the millions of people who start calling for recall votes, and will notice if elections are rigged or suspended? How the hell do you stop captain random of her majesty's navy from sending in a few marines into London to grab an emergency broadcast system and alert the country when he is ordered to do something illegal? How many independent reporters do you have to mind wipe every day before the news gets out? One way or another, it will, and when it does the country will become ungovernable. Wizards don't have the numbers to have boots on the ground keeping order. Hell, they don't even have enough boots on the ground to have a wizard per police precinct. If they are lucky, they might have enough wizards to have one trained auror per major city, and that's just not going to cut it. It doesn't matter if every single auror has all the bells and whistles that makes them a super wizard. They don't have the numbers to control cities of millions.
Even if the wizards start burning cities, how many cities can they burn a day? Even in a best case scenario, what, maybe 100 cities per day? There are more than 4000 cities around the world with populations larger than 150k alone. That's around the ballpark of the wizard's total global population. Even if wizards were burning each city to the ground, they wouldn't be able to get them all. And if wizards are being sent out to burn cities to the ground, how long until some of them stop coming back? How long before a policeman or a soldier gets a lucky shot in? How long before aurors start thinking 'I didn't sign up to commit genocide'? How long before muggleborn wizards start helping the muggles fight back? The wizarding world may be powerful, but they do not have the infrastructure and scale to fight a war like that without fracturing their own society and starting a civil war, which will give the muggles time to rebuild, reorient, recruit, and counterattack.
