Anon how do you "keep the draw a surprise"? You have to have a method or countries that randomly draw tough opponents will complain. Your world ranking under the current system will be as much luck as performance-based... oh wait, that's a lot like the current system.
*** Stream of consciousness for a bit here ***
(tldr: skip to the next set of asterisks)
You can straight seed teams in the final 16 by world ranking, which rewards the established powers, but that raises the question of how ranking is determined. This can also be a problem if rankings become too stagnant though, as the 16th team always draws USA in the quarters, the 8th/9th always playoff for a spot against the USA in the semi's etc. You could randomise it around the rankings by some arbitrary means (lottery balls, whatever), which makes it most likely that you will end up something approximating a straight seeding, but will shuffle the teams around a bit each time. Unless your ranking changes after each group game, this would make the "group round" irrelevant though.
You can change your ranking system so that every game counts, and so that it matters who you play, and optionally how much you beat them by. Not doing this is certainly part of the existing problem. The only component of ranking that is based on who you play is the fairly arbitrary weighting system for regional tournaments. Anyone that doesn't play in EuroBasket, including the USA, is disadvantaged to varying degrees.
16 team, single elimination is 15 games. An 8-team round-robin is 28 games. If you then play-off for gold/silver/bronze you've got 30 games. That may be a logistical problem, and will also raise the "too big a risk for star players", and potentially dilute the talent base.
*** Alright here comes my actual suggestions ***
Best system I can think of is.
1) Change rankings to be based on who you win/lose to, not final position.
A1) World Champs/Olympics are single-elimination tournaments (no group stage). The top 8 seeds get a bye in the first round.
A2) Consolation bracket games are played between eliminated teams (so that each team gets a minimum number of games, and to improve the accuracy of the rankings)
A3) Seeding for the draw is based on world ranking with some slight randomisation to prevent stagnant ranking issues.
B1) The top 8 teams are placed in two groups and play-off for seeds 1-8. (Some gamesmanship as teams try to drop from 5th to 6th to avoid the 4/5 -> 1 path)
B2) The bottom 16 teams are placed in 4 groups and play-off for seeds 9-16. (Some gamesmanship as you probably have a better draw if you finish 10th to avoid the 8/9 -> 1 path)
Both of those suggestions have problems but seem better to me than the current system.