Yes, that's the "Hollywood" version. In campaigns, GMs, myself included, penalize any commander and unit for trying to fight and command at the same time. It can't be done unless you adopt the WW2 Soviet "hen and chicks" tactic, where units just follow the commander. That tactic resulted in battles being lost, including an air battle where the Soviet commander landed his plane due to damage, and the rest of his squadron landed as well. The Japanese planes easily destroyed them on the ground.
Or literally any small unit tactics group. If you have two people on a side in a military engagement, you get one that's the guy in charge or they die very soon afterwards.
In any small unit engagement you have one guy making the shots where he is under fire and yes also firing off rounds of his own, because the reality is, if you have someone sitting safe from the engagement then they function off of incomplete information and you have to deal with communication break downs and other issues.
Given most engagements in the BT universe are 4-12 units (and most formations are lucky to see above 12 units during massive parts of the universe history), yeah, the commander is usually on the field. Even if we are talking a large group, like a battalion or regiment, There will be commanders for the battalion, company, and lance, because the reality is that the more people under your span of control, the more broad you need to give your orders. The Battalion commander tells the company to achieve Objective Y, the company commander will tell Bravo lance to so secure the LZ, but the Lance Commander will give detailed tactics since they have the most information.
Your flight example is actually a quite apt metaphor, in the modern military we still have squadron leaders and wing commanders and use them extensively, and they have operational command when deployed on mission. But just because you have a commander, doesn't mean you put a brainless peon who can't think for themselves under him.