What are the theoretical influences of the military's involvement in politics, especially on decision making and how the political system runs, that make a military-involved political system different from a system with reasonably separated powers?
In most countries the military have, be definition, the capability by force to impose any decision they make on the civilian authorities and population.
Normally the only safeguards are :
The constitutional and/or legal limits imposed on them which they respect, out of a combination of respect and/or fear.
Many countries make the soldiers swear an oath to uphold the constitution of their country and it is often seen as a first line of duty. However not all countries have an oath like that.
While using force to impose what you want is viable, it creates more problems (often worse ones) than it solves. At the very least it will create civil unrest (with rare exceptions).
The "Chain Of Command" mentality. All military systems operate some version of respecting the chain of command as a fundamental principle. From recruit to head of the armed forced you have that principle drilled into you (literally) every hour of the day. As most countries make the civilian head of state a formal military position as well and indeed many give formal senior ranks to some political offices, the principle is that military officers will be very, very reluctant to go against a deeply ingrained belief in the chain of command even if the top of that command is a civilian office holder.
Internal war (literal) within the armed forces. Even if a major portion of the armed forces decide to take power, that does not mean they will not immediately be faced by a split of the army into factions that oppose them or try and remain neutral. It's one thing to kick civilians into line, but it's quite another to find yourself being opposed by an equally well armed and trained group.
The problem with the military being involved in a civil government directly is that this is never stable. Inevitably, in this case, people are attracted to senior military positions primarily to gain both political and military power. This will lead to either a military led government that is prone to internal coup-de-tat and/or a civilian government that while notionally in control has to constantly "respect" the political views of the military - i.e. the military don't actually do anything directly, but they are your primary concern when making decisions.
Examples of this instability go all the way back to the Roman Empire to the present day.
The problem with military involvement in the present day is that it presents economic problems for a military-controlled ("heavily influenced") government. Quite apart from the need to pump money and resources into the military to keep them happy, the external economic world will generally be deterred from investing in your country. Economically military-controlled countries are seen as fundamentally unstable and the military are not generally focused on key issues for economic stability, like housing, food, health care, transport, industrial and financial development. On top of that a country led by a military controlled government is going to be faced with economic sanctions from major trading blocs that definitely do want to discourage military groups in their own nations from trying to take over. Economic instability (or decline) is a problem that most military officers are not equipped mentally to deal with. They are not generally economists, do not think about the practicalities of running business (in particular the bread and butter small business that are fundamental to most economies) and are not at all equipped to deal with the psychology of civilian workplaces required to make a business operate properly.
Externally there are potential issues for diplomacy. A mindset that rules by force is not generally suited to having to negotiate with countries it cannot control by force and which are not controlled by force themselves. Military mindsets that come to power by force are simply not those capable of compromise and diplomacy is all about compromise.
The final issue is that if you take control by force (implied or explicit) then you have the "tiger by the ears" dilemma. Even if a military takes over with the honest intention of releasing control "when things are stabilized" (or similar), this is a problem. If you let go too early then you might face the wrath of the groups you came to power to suppress. If you hang on too long you become dictators and the focus for all civilian grievances - the entire military are seen by civilians are one group, not different political parties with different views you can support.
Fundamentally if a military takes over by force it is because they do not trust the civilians to control things the way they want to. That mind set does not go away when you have power, so it drives the "tiger-by-the-ears" rationalization for never releasing power. Civilians screwed it all up and we had to fix it, so it is never "safe" to give the civilians back power.
But even if a military does release power this does not mean they get off the political bus. They may not be driving, but they become a back seat driver. It is very hard to ignore a back seat driver when you can hear them loading their gun in the back seat.
So the fundamental problem of military led or influenced governments is that they are unstable socially, politically and economically.