I think that all the exceptions prove the rule. Be there early!
However, there is one point that was only briefly touched on my one person - security.
I am a regular at LTN and of EZY, so know the system and fastest line between check-in and gate!
Let us say that check-in say, "OK, you have no hold bags and you can run for it." THEN security are concerned, you are flustered. Maybe sweating heavily? This is a classic of someone trying to beat the system by being deliberately late and blustering their way through.
They might choose to open your hand case and go through it in minute detail. They might choose to take you aside and question you further.
What happens to the, "It'll only take two minute to run to the gate." then?
I am one who likes to get to check-in very early but I too have been held up by traffic on M25 and had to run for it through LHR. It happens to all of us.
As for the offer of flight the next morning. If this is standard policy, it would be offered. If it isn't, it wouldn't!!