I thought it would be interesting to take a look at how the unanimous (or near-unanimous) #1 picks ended up in their respective draft class (I did go by memory so perhaps some of them weren't actually near-unanimous). I went back to look at the drafts from the mid-90's on because those drafts have players still playing.
96 - Iverson
97 - Duncan
00 - Martin
02 - Ming
03 - James
04 - Howard
07 - Oden
08 - Rose
09 - Griffin
10 - Wall
11 - Irving
12 - Davis
14 - Wiggins
15 - Towns
Now if you were to re-draft all of those drafts based on what we know now, only Iverson and Oden would not be the #1 pick in that draft. Iverson would probably go #2 to Kobe, who was just at the start of the high school era. Oden obviously would fall a great deal as a result of his injuries, but the few games he played relatively healthy definitely showed the potential to live up to the #1 pick ahead of Durant.
It is arguable that Rose (injury and strong competition from Westbrook and Love) and Griffin (Curry and Harden) might not still go #1, but I think the general consensus would be they would.
The reason I bring this is up, is two-fold. One, it proves historically that when all the experts have the basically the same player as the #1 pick, that player by and large ends up being the best player in the draft and if you are the best player in your draft you are at a minimum going to have a very nice career, but. Two, just because a player is the best player in a draft, does not mean the player is going to be a historically great player. The next Lebron James, might just end up as the next Kenyon Martin instead. Nothing wrong with Kenyon Martin, he just isn't Lebron James.
Also of note, the years where there wasn't a near-unanimous player at the top of the draft, the drafts are all over the place, but generally speaking the player that went #1 would not go #1 in a re-draft. In fact, in many of those seasons, the actual #1 pick would not even be a top 5 pick in a re-draft (Anthony Bennett, Andrea Bargnani, Michael Olowokandi, and Kwame Brown - just Andrew Bogut and Joe Smith would likely be top 5 picks in the re-draft). Thus, getting the #1 pick is great if there is a near-unanimous person sitting at the top, but isn't so nice in the other years.