I need to first state that my true knowledge here is limited. I have not read their literature, just reading about the activities outside and some arguments here and there. The other aspects of the state, I do have more knowledge about. So this is some independent thinking, to make it clear.
I think Madkhalis is a new form of collective thinking (more like a trend) that formed from the investment of secular nation leaders in the Islamic world to show leniency in their operations.
Some Western scholars called it "quietist Salafism" which I think is quite apt. In a very interesting way, Madkhalism can be viewed as a mechanism for secular and nationalizing Muslim-majority countries (ruling power) to cultivate and utilize to suppress the politicization of genuine (and sometimes radical) Islamic collective will that can threaten the status quo of the interest of the secular ruling power or compete for authority.
Weirdly, Madkhalism would be the people who would stand in the way of a change if a group wanted a better Sharia-based country for the sake of the support of the ruler.
I'm not a one-dimensional about this. I reject this unwavering support for a leader, at the same time, I do agree one needs to have stability prioritized often even if the leader is not ideal. Madkhalism is the other extreme of the Khawarijs who want to break down everything that slightly disagrees with their ideology. Madkhalis are the ones who would go as far as to excuse and promote a bad leader who goes too far and undermines Islam. I am oversimplifying things, but I think in general gist of the stereotype, works enough.
The issue I have with Madkhalism is the use of the deen to support the political actions of a ruler that we know are of self-interested basis. A counter-argument to this would be; would you rather they use secular talking points? Which is valid. Because too many people are using only secular feelings in their hate for one Arab country or the other. No. In my case, I am more irritated by how they selectively use Islam to reify justification, even cloak the intentions of a leader beyond evidence or obvious intent, giving extreme charitability where there is obviously none. That to me is too much. One can make the case that a ruler is human, and if he is a Muslim we need to be charitable with them like we would be favorable to the average Muslim whenever they are flawed. The distinguishing feature here is within reasonability.
The other problem with Madkhalism is that we can never measure to what extent it is coerced and not. Usually, there are double functions. One is enthusiastic, the other is coerced and promotion (basically gains provided by the state which we might not be able to distinguish from "enthusiasm"). It is more like, one can gain legitimacy and influence by being employed by the state to channel their interest through an Islamic cloak ("legitimacy"), and on the other hand, one cannot suddenly be too critical, because then one can end up in jail or have their life uprooted completely being blackballed. In this case, Madkhalism is too connected, if not an extension of the ruling elites.
Madkhalism outside the affairs of the state is usually no different than any other Muslims and should get their credit there since they're not extremists in their day-to-day practice, as much as you give other people credit for furthering the sunnah. But they seem to be a loose pseudo-political movement organized mostly by ideology. A specialized order within Salafism that conveniently serves the ruler well.
Another angle to look at it from is Madkahlism's parallel with nationalism. Besides the clear differences, we have one notably weird one where, often nationalism has an ideology that shapes a-priori, while Madkhalism is defined by the status quo to then justify that. The state already has an Islamic element informed by the scholarships and the traditions cultivated throughout the generations that lends much coherence of not making the countries non-secular. So the Madkhalis are likely not the main body of Islamic adherence in those countries, they are merely a specialized one, given the influence to shift public discourse. And the general non-Madkhali scholarship and tradition usually are apolitical but still fill the daily citizen with character. That is why you find plenty of pious people of pure hearts in the peninsula who are very religious despite their love for the rulers. This notion that all the Arabs there are monsters is simply not true. You might find the best people deep in some Arabian mountain valley.