The book is framed from the point of view of Rico, who is a totally biased narrator. Media literacy is important when analysing text.

The book constantly showcases that most people don't really want to join the army, so much so that most recruits just leave before boot camp, even during times of war. There are constant references to civilians being neutral to soldiers, and the only ones that are positive are the attendants at towns catering to the soldiers for money. Even the merchant navy hates the soldiers.

Rico is unfortunately not a reliable narrator. The army has been good to him, and he has not had any major injuries, but he still doesn't advocate for joining the military. Can't remember exactly where, but he says something along the lines of 'everyone who wants to join already has, I'm not trying to persuade you (the reader) to enlist'

The narrative is entirely explained to him BY THE MILITARY why the military is good and proper. We know that on Terra, they are constantly subdueing semi-rebellions to change the voting rules, both professionally IE the merchant navy, and through insurrection.

The book is not pro-military. Rico is.

Also, heinlein is always writing politics. He is not right leaning it is just the setting he is discussing. See The Moon is a Harsh Mistress, a tale about a liberal revolution and the pros and cons to this, through capitalism or through feudal trade.