I find this premise to be silly. Boston doesn't have James, Durant, Paul, Davis, Harden, Griffin, Curry, etc. Boston doesn't even have a 2nd tier star player like a Horford, Bosh, Love, Leonard, Rose, Wall, Irving, Aldridge, etc. Boston might not even have a 3rd tier star player like Butler, Noah, Teague, etc. Boston has no young guys with that sort of potential either i.e. there is no Wiggins, Parker, Noel, etc.
It is nice to have a lot of draft picks, but none of them are top 5 or even a real great shot at top 5 (heck none of them may end even being a top 10 pick). It is also nice to not have albatross contracts, but I find Aldridge's statement to be preposterous. I'd much rather be any number of teams that has at least one, if not multiple, top tier players or players that could become top tier players. Boston doesn't have that.
The last time a team had this many 1st round picks, the Seattle Sonics turned Oklahoma City Thunder ended up with top 5 picks in 3 consecutive drafts (and heck they started that run with 2 top 5 picks in the first draft after trading Allen for the 5th pick). That is how you build your base through the draft and if you want to trade the picks, that is a heck of a lot easier if they are top 5 picks as well (you know like Boston acquiring Allen for the 5th pick). Middle of the 1st picks are a nice way to find role players, but they are not a good way to build a franchise. The value from trades and/or drafted players just isn't there from a historical perspective.