They were moved back to the run position 10 seconds after being switched off, and the engines were in the very early stages of restarting by the time of the crash. It was too late.

at least one of the pilots did. according to the preliminary report, the switches were only in the cutoff position for 10 seconds before being switched back to the run position and the engines started to spin up again

Yes, and it restarts the engines, but it takes on the order of seconds; too long at that altitude. One of the pilots did that, but it was too late.

More like 30 seconds. Just throttling an already running engine up from idle (which is quite a bit above zero throttle in most respects) takes seconds.

Turbines take a while to spin up again, it's not like start/stop in a car.

In older turbine aircraft this would cause a hot start or worse. It would be interesting to know what the FADEC systems do in this case.