I lived through the 1965 blackout. (I was a kid finishing up my paper route, and I noticed a weird glow from the hallway incandescent lights of the last few garden apartments on my route, but initially had no idea why. Then I rode home in near darkness, though fortunately there was just enough ambient light to navigate by, aided by the occasional car headlights.)
WABC back then still used records and turntables, and what you hear in the Dan Ingram clip is the TT motors slowing down as the line voltage fluctuates. But all their spots, jingles, etc. were carted, and the cart drives -- at least the model WABC used -- were built with hysteresis synchronous motors, which were less prone to wow and flutter from voltage fluctuations, so they rode out more of the early "brownout" phase of the blackout. But as the voltage went to zero, eventually everything died. WABC, to my recollection, did not have a generator in their studios at 1924 Broadway(?) on Manhattan's Upper West Side. Some months later, when they moved to the new ABC headquarters building at 1330 Avenue of the Americas, there would be a standby power.
By a quirk of the power grid, the blackout did not hit New Jersey, nor Staten Island (NYC's "fifth borough"), and most of the AM transmitters were in Jersey. So WABC remained on the air from Lodi NJ with it's 50 KW flamethrower and no programming to air. Reportedly, Ingram grabbed an armful of records and drove out to Lodi, where he broadcast through the night from an auxilliary studio that was either a standby or a cobbled-together jury-rig. But WABC was able to push through, getting updates from their newsroom by old-fashioned POTS lines.
Back in those days, FM was a ratings nonentity, so it probably didn't matter much that the buildings that housed transmitters (primarily the Empire State Building) were out and didn't have backup power. The other two network O&O's, WCBS and WNBC, both 50KW I-A's, which shared a diplexed tower on a tiny island off the Bronx coast (High Island, adjacent to City Island, if anyone cares), would have also lost Con Edison power. Not sure if they had a backup generator in 1965, but if not, they surely got equipped soon thereafter.
That left WMCA, WOR, WNYC, WPAT, WINS, WHN, WNEW, WQXR and WWRL. WQXR, WWRL and WNYC had their transmitters in various spots in Queens across the river, so they probably were also out. The rest were in various locations in Jersey, so their transmitters remained hot. But the only one of those stations I recall listening to that night was WMCA, which had a backup generator at their facility at 415 Madison Avenue, and after a short pause to get it fired up, ran through the night providing news, info and the occasional record, initially with "Good Guy" Dan Daniel hosting the emergency coverage. Over time other DJ's, news anchors and reporters, and talk show host Barry Gray joined the continuing coverage until power came back in the early morning.
WOR and WNEW also had fully staffed newsrooms, and I recall them eventually getting back on air to also provide coverage. But as a kid, WMCA and WABC were my go-to stations, so those are where I tuned that night and can speak to from memory. (Interesting note, 1010 WINS had changed format from Top40 R&R to All-News just eight months earlier, and the blackout was their first big test. They must have also had emergency power, because they were among the stations later lauded for their coverage.)
(I can't talk about either of the other blackouts. In 1977, I'd just taken off from JFK Airport for a business trip to San Francisco, so I was in a news cocoon when it happened and only found out when I got in my rental car at SFO and tuned to KFRC, and heard their evening jock mention it. By 2003, I had been living in the Bay Area for years. As to Sandy, Scott Fybush's writeup is your go-to source.)