Where did the misconceptions come from?
I think that the greatest influences on this strength and flexibility conflict are the two groups that are put at the pinnacle of these two qualities, as well as a misunderstanding of what strength is.
There are two layers to the misconception that strength inhibits flexibility. The first is that muscle mass = strength, because of this the public put bodybuilders as the strongest group of people. While a lot of bodybuilders are incredibly strong, their main concern is building muscle, so strength and flexibility are rarely the main goals of a competitive bodybuilder and so relative to their bodyweight bodybuilders aren’t as strong as strength athletes such as strongman competitors or powerlifters, nor are they typically flexible, but damn do they look good.
There are some contemporary examples of flexible bodybuilders though, one of them being Jujimufu, though he exists as a bit of an outlier, as he’s had his flexibility for a longer amount of time than he’s been bodybuilding. The classic bodybuilders in Arnold Schwarzenegger’s era were slightly more concerned with building flexibility though, mostly so they could train the muscles through their full range of motion and increase hypertrophy.
Powerlifters and Strongmen are pound for pound among the strongest individuals on the planet and rarely express large degrees of flexibility, so some would argue lends credence to the idea that you can’t be strong and flexible, but this is the problem with looking at individuals at either end of the bell curve and the specific physical adaptations they have for their sports.
Powerlifters have 3 movements in which they compete: Bench, Squat and Deadlift. There are form constraints on these exercises which constitute a good and a missed lift, because of this, powerlifters limit their range so that they can achieve the standards for each exercise but no more. By limiting the range in which they move they ensure that they maximize the amount of force they can produce and lift heavier weights. If they increase their available range beyond this it means that they waste energy, as you don’t get extra points for a very deep squat.