Last season the Edmonton Oilers were a bad team. By the last month of the season there was very little chance that they would be able to make the playoffs but the fans of the team still hoped for them to win every single game. This was at least partly due to the fact that they did not have their own first round pick. This made for a much more enjoyable fan experience than the year before when the Oilers were tanking after the trade of Ryan Smyth. I don't blame fans (or management) for hoping that their team loses once they've fallen out of the playoff race since that's how the league is set up: either be very good or be very bad. This, to me, is the problem.
The reward of the first overall pick should not go to a team that has carefully managed to consistently lose. That is foolishness and is contrary to the competitive nature of sport. As much as players are trained to win, it's difficult to imagine that they're so unaware of game theory that they aren't aware it's better for the long-term interests of the team that they lose. I would imagine that overcoming those feelings is a pretty daunting task and that some players have their performance negatively effected. I'm not saying that they're throwing games as they obviously have a personal (monetary) interest in performing as well as they can, but it would still be a difficult feeling to overcome, especially if they believed that management was hoping for them to fail.
I would propose that the team finishing with the most points but still missing the playoffs is more deserving of the first overall selection in the draft. They have demonstrated that they are trying to win, but are still in need of help in order to catch the really good teams that are finishing first or second in the conference. There is a significant monetary incentive for both player and team to make the playoffs, so I wouldn't be very worried about teams throwing games in order to get a better pick, especially if the weighted lottery didn't give them a particulary good chance at success. I would advocate for a 105 ball lottery where the number of balls would depend on the team's finish in the regular season. The team with the most points that missed the playoffs would get fourteen balls (13% chance). The second best team, thirteen, and so on down the line. Eliminate the rule where a team can only advance or fall so many spots and then televise the draw (maybe do the envelope reavealing thing that the NBA does). For the second round, go back to the regular order where the team that finishes last gets the first (31st) pick.
This system would prevent tanking and would let fans hope for their team to win no matter the circumstances, which is the way that I think both sports and fandom ought to be.