My thoughts have been spelt out in groupme, but for clarity, here they are.
I strongly believe that it's not broken so we shouldn't try to fix it.
Put a constant timer on the draft, moving from 24hrs per pick for the first 2 rnds, to 12hrs for rnds 3-5 and 8hrs to rnds 6-8 AND, the most important part, as soon as you have picked send a group email to ALL GMs saying "Team X is on the clock". We then set a draft MUST be finished by day X, and just auto all picks not made, you should then see guys pick up their game next draft. Remember that every RL week is only 3 sims, so having a draft deadline of 14 days after the regular season ends (or after the playoffs end) is not very many 'missed' sims.
We should then have the draft done we with is suggested above and it doesn't have to move. It's the responsibilities of each GM to think of his other GMs, I know that I, being in Australia, try as much as I can to set a list of at least as many players I'll need if the draft is likely to get to me before I go to sleep, so that the draft doesn't hold up for others - it's simple draft etiquette, if you're unlikely to be available, set a list - it doesn't have to be 20 players deep.
Also, by moving the draft and having a lottery there is NO fair way, just a random way of distributing the picks.