Years ago, especially in midtown, the high profile boxes, a DC was assigned on the box, i.e. Rock Center, U.N., some boxes on 5th avenue, etc.
As Atlas reminds us, in the late 60s and into the 70s, it was not uncommon for the Deputy to be the only chief available (sometimes in a boro) to respond to a working fire or an incident where a chief was needed. A company that had responded on a box without a chief assigned (which was common on busy days or nights) would transmit a 10-30 or 10-75 and the dispatcher would start a roll-call looking for an available chief. Busy nights would hear dispatchers announce "Is there any chief available for a working fire in the Boro of -----?" I responded with the 43rd Battalion to a job in Flatbush - for a working fire in a store - and no closer Brooklyn chief became available the entire run. The fire was a 2nd alarm - and there was also no Deputy available at any time the fire was in progress. There were several other multiple alarms in progress. And Coney Island was a busy place itself.
It was typical for a Battalion Chief to respond to a job and not know the units that had been assigned (no tickets back then, companies were assigned by voice alarm and/or radio, companies were relocated, companies were interchanged, companies had 2nd sections, rigs were often old unmarked spares). Chiefs would arrive at a job and start finding out on the Handi Talkie by asking "Battalion 41 to the 1st due engine or truck, where are you?" Boxes would come in for the same location later in the night and there would be different companies assigned due to availability.
The Department organized new Divisions and Battalions to address the shortage of chiefs for firefighting in the early 1970s. Battalion 2nd sections did not work, did not provide flexibility. Innovations like a floating 60th Battalion in Brooklyn were tried. It was just a crazy time.
So yes, Deputies had fires by themselves. Atlas knows how busy the War Years were - the best of times and the worst of times.