Absolutely correct. In the real world, you'd have a dozen squad cars on hand to secure the scene before EMS and FD would even be allowed down the block. And besides, PD are often dispatched to unknown type rescue jobs, and *because* it's an unknown type rescue, FD would be toned out as a matter of course for possible forced entry.
Which also brings me to the house fire. "51 get some water on that fire." Right-o chief. Except I didn't even see the engine company in the establishing shot. And even if it was there, it would take a four man crew about two minutes to take a hydrant, hook up a supply line grab an inch-and-three-quarters (or two) and at least start a basic attack. A job roiling like that would have gotten an immediate 2nd Alarm to boot - and their first alarm would be a minimum of 3 or 4 engines, 2 or 3 trucks, a squad and at least a couple of senior officers. Now roughly double that for a Deuce :). But at the very least, could they not even afford to show one single engine company from their own house puttin' the wet stuff on the red stuff? And for that matter, with a fire that bad, and only a single truck company they're setting up something Really Effin' Bad by not venting the roof. Which is why there are multiple truck companies dispatched to a working fire :)
For the sake of TV production I get it, of course, but they *could* try just a *wee* bit harder?? Maybe??
And that brings me back to Emergency! from back in the day, when it was the other way around. Their house had the starring Paramedic/squad and an engine company. So many of the calls they ran could have ended very simply, in a fraction of the time, with one or two truck companies--especially the resident Tower Ladder <grin>.
The unjust dismissal lawsuits that could emanate from such a case would reverberate through the department--and the local media--for years. Not just that, but in what universe does a Battalion Chief make hiring/firing decisions in the first place? Ain't no way our Problem Child--er-Chile gets summarily canned just like that. I could see Boden giving her three written warnings then calling in higher-ups to intervene, but, like the OP stated, EAPs and the like exist for a very good reason. She sure as shit wouldn't be the first and won't be the last member of the service to drown her sorrows on company time.
So that now brings up the rest of the season. Obviously she's coming back at some point--it's network TV ferheavensakes--but what hot guy or gal will fill in for her until that happens? I haven't seen the previews, but since we're in sweeps, you can bet it won't be Joe or Jane Shlubotnik in the ambo.