So let's go over the evidence that proves these two myth unambiguously false.
Does Strength Make You Slow?
This idea is so false that when I mentioned it to my wife, she said, "that's just silly." And it is. Strength is the ability to produce Force. Force is defined as mass times acceleration. So, if my body mass and sword mass remain the same then increasing force means increased acceleration. This is just definition of terms and the Laws of Physics.
But that's not all the evidence. I have trained track & field athletes at Boston University Strength & Conditioning, one of the best S&C facilities. The track athletes there train heavy. They perform basic strength exercises like the squat and bench press in the 5-8 RM range. This approach to training is obvious just from looking at the athletes at a track meet. They have strong, well-defined muscles. They are not skinny like the cross-country team.
This is Usain Bolt - Fastest Human on Earth
One of my instructors at UMass was friends with Ben Johnson's strength coach. When Ben Johnson was the fastest man on earth he was squatting 600 lbs.
Across all of the track events speed and acceleration are critical. Some events focus on lower body power and others require full body power. But all of them benefit from serious strength training.
Squat strength is well correlated with many measures of speed, power and performance. Increased squat weight leads to higher jumps, faster running times and better performance on the field (or ice, or court) in a wide variety of sports. The number of studies showing a connection between strength and performance measures is gigantic and the results are consistent. There is no evidence to cause doubt on the matter.
Serious strength training did not used to be commonplace in most sports. After WWII, the Easter Bloc countries did research on the subject and concluded that strength training improved athletic performance. They then implemented rigorous strength training programs amongst their Olympic athletes. After this training the Eastern European countries surged ahead in medals at the Olympics across a wide variety of sports. It was only when Western nations started to follow suit that the gap closed. Simply put: it is a matter of historical fact that strength training improves performance, and that it does so in ways that include improvements to speed.
The Other Side of the CoinIt is absolutely the case that strength alone does not determine speed. The best speed is achieved from smooth, well-coordinated actions with perfectly efficient form. But perfect form is only part of the equation. A complete approach requires strength and form.
Does Strength Make You Stiff?
Specifically, does it cause a loss of range of motion? No. Strength training actually improves range of motion. Numerous studies have shown that strength training will improve range of motion. These studies have been conducted with a wide variety of populations and show consistent results.
Results have been specific. That is in studies which had subjects train upper-body pushing motions but not pulling motions, the subjects increased range of motion forwards but not backwards.
The increase in range of motion is consistent all the way up through the highest levels of training for sports. A study conducted at the 1996 Olympics found that the only athletes more flexible than the weightlifters were the gymnasts.
Properly coached strength training will even specify an increase in range of motion. There are exercises that can only be done if the athlete increases their range of motion over what a typical sedentary person starts with.
The Other Side of the CoinWhen most people are concerned about loss of flexibility they are imagining the the sort of bodybuilder types they have seen who look like they are inflexible. But those people aren't training for strength. They aren't training the way athletes train. They use the same tools, and in similar manners, but it's not actually the same activity.
Yes, it is possible for strength training to lead to loss of range of motion. But preventing this is so easy that it is barely necessary to mention it when talking to martial artists.
How to Prevent Stiffness from Strength Training
1) Stretch. A few times a week, stretch all your joints. When you think about the "meathead" bodybuilder, you don't see them in the stretching part of the gym. Whereas the typical person doing martial arts is going to stretch and may even stretch more than they should.
2) Full Range of Motion Exercises. Most reps on most exercises should include full range of motion. It is not actually necessary to hit full ROM on every single exercise. In fact there exercises where you shouldn't such as the Power Clean.
That being said full ROM should be the norm in strength training. And when you reach a point where you can't hit the full ROM then you've found your limit (for the day) on that exercise. Don't do the last rep with incomplete ROM. And if you can't do full motion at a given weight then the weight is too big.
There is one other aspect of loss of motion from strength training. It's possible for a muscle to get so big that some range is lost. A typical NFL lineman can't touch their elbows together in front because their pecs are so big. But most people simply don't have the genetic capacity to reach that point. And these losses seldom, if ever, limit health or performance. The lineman does not need to be able to touch their elbows together.