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You don't count the slips of paper in the hat. Instead you count the number of people who put in slips of paper.

That will give an average, but it won't give the distribution (min/max/etc)

I believe the parent meant breaking it down as "if you earn $100k, add 10 slips that each have $10k on them". The number of slips will be more than the number of people in the room, but that's fine. It's less practical than the "seed + pass round" method that someone else recommended imo (due to handwriting notability), but still works in a more technical sense.

You're right, you can't, I was looking at the problem posed by grandparent comment of getting only the average.