So anyhow, we reached there 2 min prior but were advised that the flight already left and the gate was closed.

Say the flight "departure time" is 10:40:00

You simply can't arrive very close to that time. You have to arrive before the "gate closes".

That's that.

The time the gate closes is indeed given on the boarding pass stub, and elsewhere in many places.

It's usually ten, twenty or thirty minutes before the stated departure time. Perhaps 10:20:00 in the example.

Regarding your four questions:

1. You checked in, but, did not make it to the gate before the gate closed. This is totally commonplace. (Indeed, I'd guess that most folks who don't make it to the gate in time, were already checked in.)

2. "Departure time" is neither the time the aircraft leaves the gate nor take-off time. So there's nothing to "confirm". "Departure time" is a nominal time. The gate closes at a certain time (10:20 in the example) and that's that. Everything after that simply depends on airport operations.

3. You can get that info on the various tracking sites, eg flightaware.com. But it's totally irrelevant because (sorry for the bad luck) you did not make it to the gate by the "time the gate closes".

(Interestingly, I've noticed that many/most boarding passes now only display the gate closing time, rather than displaying the "departure time". I guess this is sensible but I find it confusing, personally!)

So anyhow, we reached there 2 min prior but were advised that the flight already left and the gate was closed.

Say the flight "departure time" is 10:40:00

You simply can't arrive very close to that time. You have to arrive before the "gate closes".

That's that.

The time the gate closes is indeed given on the boarding pass stub, and elsewhere in many places.

It's usually ten, twenty or thirty minutes before the stated departure time. Perhaps 10:20:00 in the example.

Regarding your four questions:

1. You checked in, but, did not make it to the gate before the gate closed. This is totally commonplace. (Indeed, I'd guess that most folks who don't make it to the gate in time, were already checked in.)

2. "Departure time" is neither literally the time the aircraft leaves the gate nor take-off time. So there's nothing to "confirm". "Departure time" is a nominal time. {Note too that FWIW, the airbridge moving away, doors being locked etc, are all different from literally "the wheels beginning to roll".} The gate closes at a certain time (10:20 in the example) and that's that. Everything after that simply depends on airport operations.

3. You can get that info on the various tracking sites, eg flightaware.com. But it's totally irrelevant because (sorry for the bad luck) you did not make it to the gate by the "time the gate closes".

(Interestingly, I've noticed that many/most boarding passes now only display the gate closing time, rather than displaying the "departure time". I guess this is sensible but I find it confusing, personally!)