What would cause a car to die completely even after a jump-start...?

Battery.

That is probably an elusive problem seen in hot environments all the time but less often in cold environments. When I lived in Phoenix as many as half my batteries died this way. An internal connection inside the battery has broken, usually where the positive post connects to the plate. Current going through the battery will temporarily weld part of the connection back together and bring the battery back to life, which makes them almost always test good. Coming back to life is a good indicator - there are not many other causes of that - but you should get a solid diagnosis. You can do it yourself with a voltmeter when the car won't run. Measure the voltage on the battery posts (not the clamps, to be sure you are measuring the battery itself). You will see a voltage well below 12 volts... usually 8-10 volts. First test: turn on the headlights and watch the voltage. If the voltage rises while the lights are on that is a solid diagnosis, but sometimes the voltage does not rise with that little current. Second test if the first gives no results - with the lights still turned on, whack the positive post with a hammer, about like you are cracking a Brazil nut... firm but not so hard the case breaks. If the voltage rises sharply when you hit the post you have confirmation. The hammer trick will also get the car going every time in an emergency when it has a bad internal connection, but sometimes it requires a lot of whacks - as many as a couple dozen in my experience.

This sort of failure first started appearing in the 1980s, as near as I can tell. It was originally associated with side post batteries that made their first appearance around then, and there was a good reason to blame them. Side post batteries have connections that come off the plates sideways. The underlying reason is the calcium grid battery chemistry that became popular around the same time, allowing low and zero maintenance batteries to be sold. Calcium alloyed grids have a lower self-discharge rate and lower gassing than antimony alloyed batteries, but they are not as tolerant of deep discharge (like leaving the lights on) and have a quirk: if chronically overcharged the positive plates oxidize and grow in length and width. In hot environments the regulator in most cars fails to compensate for the actual battery temperature and the battery is chronically overcharged. The first source has more on this.

Anyway, if a broken internal connection is your problem you need a new battery.

Didn't you ask this question a while ago? Did you check the part(s) everybody suggested you check?