What I'd like D&D Next to take away from BECMI / BD&D:
The idea that you can start a campaign with minimal rules and options and increase your options slowly to the give the players and DM time to master the game as they go along.
BECMI defined itself by creating tiers of rules complexity, not just tiers of character levels. Basic, Expert, Companion, Master, and Immortal slowly added in more detail.
My thinking is that combat isn't slow in 3rd and 4th editions just because of the thousands of pages of rules or because of the grid, it's because of a lack of player and DM mastery of the rules. We have too many options too quickly, particularly in the DDI character builder.
I am willing to bet that no matter how streamlined the new rules set is, people trying it out for the first time will complain that it "runs slow" because they will try to incorporate too many rules at once. Even experienced players and DM's (and perhaps especially those folks) will do this.
Modules will not only hopefully allow this gradual learning curve, but will encourage it. The same goes for the raw number of classes, races, feats, etc. The volume of new material needs to be controlled so that we're not drinking from a fire hose.