The first thing is to avoid equating Roman ranks and positions with modern offices. There's a trend to equate centurions with NCO's and attribute them as the reason for the army's success. We're projecting the general disdain for newly minted officers of our modern age.

The centurion is not an NCO. He was an officer, albeit a lower ranking one. In the Middle Republic they were elected by the soldiers (specifically the hastati, principes, and triarii). The centurions in turn then appoint their second-in-command optiones. This election means it could go to the most popular, the most skilled, or simply the one with the most auctoritas.

Under the empire centurion promotions were personally approved by the emperor. These weren't simply grizzled veterans rising through the ranks based on merit alone - the emperor got himself involved because they were from a politically significant class and one he wished to claim patronage over. Goldsworthy severely doubts the idea that centurions would rise up in rank starting in the 10th cohort and moving upwards until the prestigious 1st cohort - this would have taken a ridiculous amount of time or else each centurion would only spend a few months serving with a cohort before moving ahead. Scholars had generally believed most were promoted from the ranks but there is actually little evidence to support that. The majority of inscriptions by centurions make no mention of their rise up the ranks.

It was possible to go from an ordinary soldier but this took considerable time (15-20yrs) and required considerable literacy. More likely was a direct commission from the equestrian order or, if they were less wealthy, appointment following service as a town magistrate. The office of centurion had considerable authority, they might be called to administer regions of a province and would be the senior representative of the Roman state. A Primus Pilus was even sent as ambassador to Parthia.

As for senior officers, they would not have been 'green'. The Romans didn't have OCS or Sandhurst, they would have learned through observation - on the staff of their father, or brother, or uncles as senior officers on campaign. They'd have been trained in oratory from birth and would have been expected to lead. When old enough they'd stand for election for junior officer roles such as the quaestor (essentially a treasurer). By the time he became a tribune he would have had experience in the field, not in a leadership role but would have observed how to do so. This didn't make them hardened vets - the tribunes in Caesar's army started panicking out of fear from fighting the Germans, but then pretty much everyone was scared of them(except Caesar). But these weren't bumbling lieutenants who'd never seen a battlefield before.

We tend to downplay the average officer because of our beliefs in meritocracy and trained officers and these noble politicians 'playing general' offends our belief in the separation of civil and military affairs. But men like Caesar, Scipio Africanus, Claudius Marcellus, Agrippa, Agricola, etc... all learned their craft this way.