#1 overall picks since 1980 who have been on a world championship team: Mark Aguirre (twice), James Worthy (thrice), Hakeen Olajuwon (twice), David Robinson (twice), Shaquille O'Neal (four times), Glenn Robinson (once), Tim Duncan (four times). Worthy and Duncan are the only players on that list to get a ring in their first seven seasons. If we exclude the last seven drafts, then 28% of first overall picks since 1980 have won an NBA title.
I think you're looking at this the wrong way, because it's not #1 pick or nothing. When many people believed teams like the C's were tanking in '07, it wasn't for the #1 pick, but for #1
or #2 pick. When teams tank, sure they hope for the #1 pick, but they also think other top picks are a great consolation prize as well.
And while an NBA championship is the ultimate goal you're going for, if you broaden the guidelines and consider teams that made the NBA Finals the picture is a little different. (I would actually prefer considering teams that made the conference finals, as I figure the vast majority of the time, teams that make the conference finals are considered contenders, and really what you want is for your player to make you a contender, but that would take more work to get those numbers...)
Like you did going back to 1980, but looking at the top 3 picks and teams that made the finals (instead of #1 pick and championship) there were 81 players taken in the 27 years from 1980-2006, and there have been 82 Finals appearances by those players, and 42 championships.
So your math was seven #1 picks in 25 years have won championships 7/25 = 28%
But those seven players accounted for 18 championships (though 3 were duplicates due to multiple #1's on the Spurs teams). So 15/31 = 48% of the championships since 1980 have been won by a #1 pick. Looks a little better.
Now for my additional parameters (#1-3 picks, NBA Finals):
The draft classes starting from 1980 on (excluding the last 7 years like you did) have had a player selected #1-3 make the NBA Finals from every draft class except the 1986, 1998, 1999, and 2002 classes, so 21/25 gives teams an 84% chance of taking a guy who will make the NBA Finals.
Scaling it back to just #1 picks, 15 of the 25 since 1980 have made the NBA Finals, that's a 60% chance of getting to the Finals. I'll tank for that.
Draft classes since '80 with a #1-3 pick winning a championship? 14/25 = 56%
Now I will admit some of the numbers are skewed a little with top 3 picks who were scrubs on the bench of a good team (thank you Adam Morrison and Darko Milic with 3 rings between the two) or just role players (Coleman, McDyess, etc.). So here's the master list I put together (and as always there may be some errors as well):
Year Player Pick # Rings Finals
1980 McHale 3 3 5
1981 Mark Aguirre 1 2 3
1981 Isiah Thomas 2 2 3
1981 Buck Williams 3 0 2
1982 James Worthy 1 3 7
1983 Ralph Sampson 1 0 1
1983 Rodney McCray 3 1 2
1984 Hakeem Olajuwon 1 2 3
1984 Michael Jordan 3 6 6
1985 Patrick Ewing 1 0 2
1987 David Robinson 1 2 2
1988 Rick Smits 2 0 1
1988 Charles Smith 3 0 1
1989 Danny Ferry 2 1 1
1989 Sean Elliot 3 1 1
1990 Derrick Coleman 1 0 2
1990 Gary Payton 2 1 3
1991 Larry Johnson 1 0 1
1992 Shaquille O'Neal 1 4 6
1992 Alonzo Mourning 2 1 1
1993 Anfernee Hardaway 3 0 1
1994 Geln Robinson 1 1 1
1994 Jason Kidd 2 1 3
1995 McDyess 2 0 1
1995 Stackhouse 3 0 1
1996 Allen Iverson 1 0 1
1996 Marcus Camby 2 0 1
1997 Tim Duncan 1 4 4
1997 Keith Van Horn 2 0 1
1997 Chauncy Billups 3 1 2
2000 Kenyan Martin 1 0 2
2001 Tyson Chandler 2 1 1
2001 Pau Gasol 3 2 3
2003 LeBron James 1 0 2
2003 Darko Milic 2 1 2
2004 Dwight Howard 1 0 1
2006 Adam Morrison 3 2 2