Army guys XO's roll out all the time. But on that level? I think not. Not unless it is for a funeral ceremony or something. And even then it's EXTREMELY rare. No I think this game missed the actuality of it all. Lasky had no business down there and Palmers military prowess for conducting spartan missions should've stepped up. Also I think you're right in that he does have a military advisement board. The game didn't hit on that but it only makes sense that they would have.
About as far into it as they get was in the mission "Midnight" when Lasky says that higher ups didn't like how Del Rio abandoned the chief on Requiem and put Lasky in charge for the time being.
Again, not to divulge too much information as to not violate opsec but it is uncommon to see higher tiered officers with boots on ground in a combat zone. It's not a certainty I'm sure but for the most part you won't see someone of Lasky's rank riding around in a HMMV with no doors. Lol. That's just fictional entertainment. Kinda like hurt locker.
That's my experience from the Army side of the house.
EDIT : I don't mean that they aren't deployed, I mean they don't conduct routine missions and go on simple convoys. Not their style. That's what your enlisted soldiers are for. Case in point Master Chief is E9. The highest enlisted rank you can attain. This is a stretch but yes it is perfectly believable.
On that note... This is like what... 2556? I'm sure things are done a lot differently in the UNSC rather than today's military. ROE and EOF for a covenant or a flood or a forerunner is going to be much different than it would be for fallujah or Kandahar. Lol. I don't think humans in general would care too much to save any blood thirsty aliens, cold sentient robots, or space zombie abominations. Just saying.