I did an analysis of the first 5 rounds from 1965 to 1974 and found that:
11 of the 51 total picks played at least one game for the Twins
6 of the 51 picks never signed a contract with the Twins and were available in later drafts
34 of the 51 players drafted never played a Major League baseball game
The best players drafted and signed by the Twins were: Graig Nettles, Dave Goltz, Bert Blyleven, Eddie Bane, and Butch Wynegar.
During this 10 year period the Twins finished 2nd (1), 3rd (2), 4th (4), 5th (1), and 7th (2)
Two observations: 1. Eddie Bane?
2. Your summary of the Twins finishes looks more like that of the 1975 thru 1984 season since I seem to recall a number of first place finishes in the 1965 thru 1974 time frame.