Homebrewer here, they generally don't die, they just go dormant. It's quite normal to harvest yeast at the end of fermentation, to the point where many breweries do so to save money. You can even use the harvested yeast to make bread with (which is how bread used to be made)
Also homebrewer here, you're correct, that yeast can be reused several times, limited by the yeast mutating away from what you want, though many brewing yeasts have been reused for centuries
The yeast is only half done with its intended activity too. What it tries to do:
Digest sugar into poisonous ethanol (which we quite like)
Wait, trusting the ethanol to keep everything else out
Digest the ethanol to acetic acid (vinegar)
Wine vinegar is grape juice done to brewers yeast's plan