What happens when Rhylanor says, we aren't sending troops to defend Regina, screw 'em. Why would Reginan's fight to save Efate?
And what would have happend if France (or Belgium, or Netherlands, or any other NATO country, for what is worth) would have refused to defend Germany in case of a WP attack?
They would have the enemy at their gates instead of at its neighbor's gates, So I guess it's to Regina interest that Efate don't fall, and to Rhylanor's interest that Regina holds, as they are in the enemy's path.
What would be done? The Imperial military has no esprit de corps? How would the Zhodani lose? How does the Imperium stay together? These are just off the top of my head.
The Zhodani lose and the Imperium stays together because the IN is strong, not because the IA is strong.
As much stronger as Zhodani are against Imperial Army, if their fleet loses, they are stranded, either in their side of the border or in the imperial one. The situation would be as the Japanese garrisoning small islands an atolls in WWII when USN achieved naval and air superiority in the zone, they were stranded and useless, they needn't even to have cared for by the USA.
An extreme of this is a planet like Glisten. According to FFW rules, Glisen would have 500 TL 15 battalions (as much as Rhylanor, as both are pop 9, TL 15 and hostile atmosphere), but if naval superiority is achieved in the system, as long as there's a single squadron with beams in the system, Glisten will surround (so, why are those 500 battalions there? Just to make beautiful parades?).
Even so, I believe the Imperial Army exists, both due to canon references and because what I (from my lack of military experience) think is logical, but its main mission is, IMHO, internal defense, instead of external defense (as told, that is carried by the Navy).
The Imperial Army, aside from being a cadre for local armies, gives the Imperium a credible threat of intervention. While the Navy can nuke a planet until it glows or sterilize it (it IIRC has done that in the equatorial region of Ilelish after its rebellion, according to MT history), but cannot occupy it while maintaining it a productive/population center. This needs dirtside work, and I guess not all dirtside troops of the Imperium are the highly trained Marines, as not all need to be.
And for this internal work, the possibility you pointed above about local armies refusing to comply is, IMHO, more real, as they may be the next victims, and because they would have to endure the animosity of this one neighbor, so, it's beter for the Imperium to use own (Imperial) troops, and, as I said, I don't believe all are marines.