"They're sure they can get done in that amount of time, so they take it easy for the first half. Then they get progressively more panicked as the deadline approaches, and they're late anyway, and/or you ship on time ..."

I found that the more experienced/older I am, the less this happens. We even finished multiple such tasks sooner then was required - of course contributing factor was that deadlines were sane. I somewhat used to have this tendency when I was younger. Experience makes you better at managing time risks. It is actually one issue I have with cookie cutter agile. Everyone is so micromanaged by the system, that they don't get to get experience needed to learn the above.

There are also contributing cultural factors. In many companies, people who stay late last week tend to be praised and rewarded over those who work in predictable speed and kept eye on the deadline from day one (not in my current team). Incentives matter and finishing tasks at the last moment at cost of evenings/weekends makes you a hero.

I personally know people, including leads of agile teams, who believe that last moment desperate effort is somehow necessary. Like you are lazy if you dont. That means there will be last moment effort, whether it was needed or not. So if the team is going to be on time, an additional work is found to be done - or more often possible logical steps to make the deadline are not taken (a feature the customer explicitly said is not necessary is done anyway etc). I wish this would be a joke or exaggeration, but I have seen it multiple times already. While details were each time different, it really did boiled down to techies essentially organizing that desperate effort for themselves.

It does not matter what the process is, how exactly you estimate etc. As long as not doing the responsible thing is rewarded, people will do that.