Well if I were him I'd have got up, whisperd to the teacher what had happened and that I had to leave, thank the kids calmly and tell them I had to go and when they watch the news later they would understand why, start giving orders for the land borders to be closed, aircraft grounded and rouge ones intercepted etc...get my ass to a secured location and make sure the line of succesion is likewise protected in case Im taken out...
They're not totally analogous situations granted, I guess my point was: people fumble the ball occasionally. They hesitate under moments of huge pressure and sometimes make the wrong calls, even if they were deemed to be one of the best Captain's in Starfleet, or the best American to run the US.
If anything I think the years and years of having Picard virtually always make the right call were less realistic than seeing someone screw up on occasion. But while the show might establish certain Captains as almost infallible, it's certainly never established that all Captains are. To the contrary, the commanders of other vessels often played Picard's foolish foils throughout the show, there to be wrong while our hero was right.