Well one of the big aspects is that everyone is in the same boat when it comes to the +/- 4M budget situation. Yeah, I can only spend $14M on Training this year, but so can everyone else. Sure that only takes 2-3 years to overcome in a mature world, but that's $50+ before I even have the team aligned how I like it from a monetary standpoint. And in some sense, that puts me on an unlevel playing field with the owners who do have their budgets lined up as they like.
To Mike's point about making a flurry of trades, because owners do not have any attachment to certain players (I drafted this guy myself last season) they are more likely to part with players to get their organization in the direction they want (whether that's competing or rebuilding). This isn't a huge part of the appeal to me, but equating the situation to an S20 team isn't a fair comparison.
Lastly, it is just plain fun to pull the arm of the slot machine and see what it spits out at you. Huge part of the appeal for me. Of course, the larger problem with this is that some owners will look at their draw and say "screw this", then get out, which they wouldn't have done in a mature world because they would have picked the team out themselves.
I understand the drawbacks to it, but I draw a lot more pleasure out of being in a world from S1, and watching that history build. Like I said, it may be a matter of personal preference.