Shoot, maybe there is a way to combine several of these ideas. Do it like they do the all-star selections, with various inputs that would prevent any one item completely overriding the others. For instance, give equal weight (for the following admittedly brief sample list that I didn't go back and review all the options in detail for, that would be 20% each) to:
- random selection of all non-playoff teams
- straight inverse record or a lottery-based selection weighting towards worst record
- Captain's idea of GM selection
- 5 year (±) weighted selection (with the requisite late lottery "positioning" if you hit on a top 2 pick the previous year, or whatever)
- whatever other somewhat reasonable system someone comes up with if/as necessary
So, say you've been a really crappy team, you would get good (low) expected draft position values from 2 and 4. If you are truly a team that tries (not tanking), but just isn't very good, you should get good values for 3 and maybe 5. Number 1 would truly be random, obviously. Then you just average all the values and if there are ties, do a coin flip or mini-raffle for the order.
If you are the Lakers, which sneak into the lottery this year despite having LeBron and AD, and due to your stupid signing of Westbrook, you should score high (bad lottery position) on numbers 3, 4, and maybe 5, effectively eliminating you from getting a top pick.
Obviously you can play with the number of inputs (5 above, but could be whatever number someone wants to go with), expected outputs for various cases or situations, odds for the lottery-based portions, etc. But maybe something like that can bridge the gap between everyone wanting one or the other.
But don't let fans vote, by any means!