I was thinking about this some more, and I really like the idea. There does seem to be a bit of an issue keeping it balanced if the number of characters fluctuates (more characters taking stats each round dilutes the pool more quickly presumming high stats go first, and changing the number of sets rolled changes the distribution), or having to deal with generating a single character (for example, as a replacement for a dead character).
So, I was thinking that you could instead draft a whole block (six stats, already assigned to attributes), instead of just a single stat: Roll out N attribute blocks for the pool. Decide your draft order. First player picks a block from the pool, a new block is added to the pool to replace it, next player goes, and so on. Each player gets to pick from the same number N blocks, and what is left in the pool can be kept around for replacement characters who would just continue the process from where it left off.
Options for tweaking, depending on the balance of equal vs. random you want in the party:
1) players can swap one pair of stats. So you can pick the high DEX, low WIS but otherwise decent stat block for a tank cleric and just swap them.
2) the player can have one block tossed and replaced with a new one before they make their pick, giving them a chance to freshen the pool -- adds a bit more excitement, too.
3) while the blocks are random, you could assign whatever conditions are needed for a block to be added to the pool -- e.g., a minimum of one stat below 10, min/max point-buy value, min/max total score, min/max total modifier, etc. -- so that no block is crazy good or crazy bad.
If you use all three options, I think you could get by with just a pool of 3 or 4 blocks.