What would happen if you miss the bag drop-off deadline and it's kind of the airline's fault for bad ground service?

As others have already written, it might depend on the airline, ground staff, airport, terminal, maybe also your status/class of travel, etc. But it's certainly not the case that inefficient handling or the fact that airport personnel seems partly responsible for a problem makes a big difference.

I once missed a flight because of very long and unusual delays before carry-on screening (I learned afterward that many members of the security team were in training and screening began late that morning). I was at the airport long in advance and followed all instructions (was even directed to a line that turned out to be slower based on the flight I was supposed to board) but I had no luck getting help from the airline. I had to rebook myself on another flight at my own expense.

That's just one anecdote in a slightly different situation (different companies are usually in charge of security screening and check-in/ground handling) but the most important point is that airlines will not necessarily assume responsibility for long queues at the airport.

By contrast, once you are “in the system” (checked-in/baggage dropped/present at the gate) and something bad happens, it's much easier to get help in reaching your destination or at least some form of compensation (e.g. for lost luggage).

There are always extenuating circumstances; however, I once arrived quite early and it seemed there would be no problems. I had just been released from a two month stay in the hospital after having sustained a TBI (traumatic brain injury). I had just learned to walk again, so had a wheel chair waiting for me at the front. Going through the ex-ray machine, there was a delay because of a cast on my arm and they needed to leave to get a special piece of equipment. We waited at the gate for some time. I had to unexpectedly go to the bathroom which I required assistance from my sister. I had been in a comma at first in the hospital and my body was very run down. We came back to the gate and waited for the three people in front of us to go through the gate and then the flight attendant started closing the door in front of our faces as we stood there in disbelief. She said, "Sorry, but it's time.". A woman nearby saw what was happening and jumped up to give her grief about treating a "crippled girl" like that, but it made no difference. My point is that a lot of it depends on the individual staff that may have just had a bad day. I don't believe the 15 extra seconds it would have taken us to cross the threshold would have delayed the plane.

Ultimately it would depend on the ground staff working that day.

The check in rules are fairly cut & dry, you are supposed to be checked in XX minutes before departure. End of story. Most airlines recommend that you arrive much earlier than that to make sure you have time to be checked in. In your case you arrived at 75 minutes, which just barely qualifies for suggestions by Delta for domestic flights (they suggest 2 hours if it is an international flight).

If the check in counter is understaffed, then there is a chance that an agent will poll the passengers in line to see if any are tight against their departure times and perhaps move them to a priority line for check in. In a case like this there would likely be some forgiveness if you checked in a couple of minutes.

If the counter has a standard staff quota, then they will likely not have much forgiveness for late check ins.

Ultimately it is the passenger's responsibility to arrive early enough to complete check in, security procedures and present themselves at the gate for boarding at a designated time. If you are late for any of them, it is 100% your burden to bare.