This is one of the reasons why many amateur golfers still avoid joining golf-specific exercise and conditioning programs. They mistakenly link every weight-training program with muscle building and bodybuilding.
Although it is true that muscle building and body building will tend to make somebody stiff, the facts are that weight training can either be used for body-building and building of muscles, or it can be used to build strength.
The golf specific weight training programs are aimed at building strength and endurance. Not muscles. A genuine golf exercise program will have nothing in its weight training routine to remotely relate it to muscle or bodybuilding.
In fact stronger conditioned muscles will tend to dramatically improve flexibility for golf, rather than reduce it. Flabby, weak, unexercised muscles are the ones that will tend to be very stiff and over time will take away the flexibility for golf in anybody.
Stretch exercises, which work best with strengthened muscles are very effective in helping to increase flexibility. These exercises usually have a major impact on the quality of the golf swing for most players.
Therefore it is not true that incorporating dumbbells into golf specific exercises makes a golfer lose flexibility. Rather the entire program plays a major role in enhancing flexibility for golf and the results from players who have tremendously improved their game, speak for themselves.