I started writing a lengthy explanation of how armies work in Stellaris (at least as far as I remember and on the last iteration I played) because it actually has some of the same issues, fleet engagements are key and ground wars are kind of unglamorous but necessary.
However that turned out to be very long and I suspect it has far too many mechanics to fit into Starfire, unless you're expecting to write a whole game engine to handle it. I seem to have written plenty without including it.
Noting in advance that I don't know the rules for Starfire or your ability to implement big changes my proposals:
1) Raising large armies inherently takes a long time, and an army which is attempting to assault a planet needs to be large. You need drill sergeants to get recruits through basic training, you need uniforms for the recruits, you need equipment for the recruits to train with and you need equipment for each finished unit. (Side note I skimmed this article about the US mobilization in WW1 which states "the materiel side of mobilization was the most costly, complex, and time consuming."
https://history.army.mil/documents/mobpam.htm)
From your post I gather that Starfire only has 1 size of army unit, and it is only used for assault. On this basis there should be limited ability to train armies which can invade other planets.
Aurora has Ground Force Training Facilities which represents the training and equipment production facilities needed to train armies. They are big, expensive and take a lot of population (the same as a research facility). They can only build 1 army at a time so they also limit the rate at which you can build up your forces.
I don't know what your options are for adding new structures to Starfire or RPing the cost of building and maintaining them.
1a) Planets should need a specific building in order to train assault armies.
1b) That building should be relatively expensive and have an upkeep.
1c) The number of simultaneous armies trained should be limited by the number of buildings (without knowing how many armies are in a typical Starfire invasion this might be 1:1 or 10:1)
1d) (optional) Training additional armies over the normal training capacity should cost exponentially more, eg 2x cost for 1 more than capacity, 4x cost for the 2nd etc. This reflects the cost of overworking the facilities but still provides flexibility in emergencies.
2) What ability do you have to add extra unit types to Starfire?
3) Training takes time and a quickly trained army will be much less effective than an army with lots of training. An army can be deployed early with less training than desired but this impairs its ability to perform. Additionally training in 'the real world' outside of boot camp is probably more valuable than the ability to run faster through an obstacle course.
Racial stats should represent the maximum performance for an army (it sounds like special forces are too small to be represented, they would be the logical exception to a limit).
3a) The training level of the army should be reflected in the time it takes to raise the army. An army should have 1 grade per month (or other sensible timeline) it spends being raised, up to the maximum of the racial crew grade. This allows armies to be raised quickly but makes them much less effective than armies raised in advance.
3b) It may be better to have non linear returns on investment depending on how large scale of possible grades are and how much benefit you gain from each grade. For example maybe the first grade should take 2 months of training and every grade after should take 3 months.
3c) Alternative: Armies are deployed as soon as they have 1 grade. Once deployed they gain extra grades up to the racial crew grade maximum by conducting field exercises. This should takes a certain amount of time per grade and cost money to perform.
3d) Regarding your point about disbanding armies once they've conquered the target.
Once you have possession of the target planet you need some way of keeping it under control, which is presumably provided by the conquering armies initially and then they get swapped for lower quality garrison troops.
Could you make assault armies explicitly single use and have them priced on that basis?
Any assault army which is expended in this way can generate something like an Aurora Cadre and give a discount or bonus when training another army, either taking less time to train or gaining grades faster during training.
On the other hand maybe taking some or all of these steps will make the armies sufficiently valuable that they will be worth keeping.