In the earliest days of MMA competition the athletes pretty much came from a single discipline. There were guys from kickboxing gyms, guys from jiu jitsu dojos (usually Brazilian), guys from boxing schools, guys from wrestling programs, etc. The interest in MMA had a lot to do with answering the old martial arts question - if a tae kwon do guy fights a judo guy, who wins? I wouldn't argue that those contests actually answered those questions - a lot of the result has to do with the rules of the contest (I still think Brazilian Jiu Jitsu is a lot less useful if groin strikes and eye gouges are allowed, but I could be wrong) and the actual competitors, but a lot of interesting information was generated anyway.
As the sport of MMA has progressed into a big money, high stakes endeavor, most of the athletes have moved away from that single instructor or single school model. Modern MMA athletes may have a single-art background - say, they were wrestlers for many years - but they all train in a very cross disciplinary environment before reaching a high level of competition. You just never see a guy win in MMA at a high level without, for example, training for many months in BJJ.
The top MMA athletes may train at one facility or many, but they all have multiple coaches, one or more for each discipline in MMA (usually wrestling, submissions, and one or more striking arts), with often a head trainer overseeing the fight plan and training schedule and an additional strength & conditioning coach to make sure the fighter is in shape. Why? Simply because no one teacher knows enough to adequately prepare an MMA fighter for all the things that can happen in the cage. And the good teachers all know that - does anyone think Lyoto Machida's father (an accomplished karateka) gets pissed that his son took BJJ lessons? There's no way he'd be as accomplished a fighter as he is if he hadn't done that cross training.
The traditional martial artist is in a slightly different situation. For example, I personally want to become as accomplished a karate practitioner as I can be. My art includes some grappling (joint locks and throws), which I have to learn, but relatively little ground fighting, for example. If I never learn ground fighting it will be fine with me. So there is really no need for me to take BJJ or wrestling lessons. Another student from my style might have other goals, and that's fine, of course, but I couldn't say that every traditional martial artist has to cross train in different martial disciplines. You could argue that someone like me will never be a complete fighter without studying ground fighting, and maybe you'd be right, but that's okay - being a complete fighter is not my goal.
So maybe not everybody needs to cross train. But every person seeking to excel in any martial art needs to pursue a rigorous strength and conditioning program. Why? Because no matter what your art, it involves movement, and movement only happens when muscles contract. For any given level of skill, the stronger and better conditioned those muscles are the better the technique will be. Now someone with great skill may be better than someone who is stronger but with less skill, but if that person with great skill becomes stronger, they will get better than they were.
There are other benefits to good strength and conditioning. A better conditioned person will sustain fewer injuries, leading to less time away from training, which will help them improve their skill level. A better conditioned athlete will be able to train longer without fatiguing, leading to higher quality training and a higher skill level.
Strength and conditioning training can be overdone. There is an upper limit to how much hypertrophy a martial artist should have - huge biceps are probably not important to a martial artist and may be slightly counterproductive. Strength and conditioning takes time and energy, which leaves you with less time and energy for skill development. Anyone spending an hour and a half a day on strength training probably won't have anything left for becoming a better martial artists. Strength and conditioning needs to be both specific to your art and efficient in terms of time and energy expenditure in order to be beneficial. Your program has to do enough to optimize muscle strength in the right areas without compromising your skill training.
Here's the next question you should be asking: Is my instructor qualified to both teach me proper karate technique and guide my strength and conditioning program in the most effective way?
Obviously, I can't answer that question for anybody else. It is entirely possible that any particular karate instructor also happens to have the knowledge base to be an effective strength and conditioning coach. However, those areas of knowledge are distinct. Understanding good technique has very little to do with understanding those aspects of muscle physiology that inform good training practices.
Think about the NFL, or any other professional sport. All these teams have coaches to teach skills to players, but they hire a separate guy to do the strength and conditioning. Why? Because they want someone who has dedicated a lot of time to learning what there is to know about strength development.
I think that your dojo should be the same. If you have a head instructor who is up to date on the most current information on strength and conditioning, that's fantastic. But very few people have the background or time to learn about S&C and still improve their technical knowledge in their art and their ability to coach effectively. Instead, your instructor should have a S&C person either on staff or consulting with them to guide your strength development program. That person should have input into class design (how much time should be allocated to different parts of training), warmup design, assessments of new students, exercise choice (not choosing skill development drills, but choosing schemes for conditioning exercises within the class) and so forth.
Just to complicate things more, your S&C guy (or gal) also needs a fairly extensive knowledge of your art. Ideally, that person would be proficient in your art, but if not, they need to spend considerable time studying the way you move so that the program design can be appropriate. This is no different from how S&C coaches work in any other field - guys who design programs for golfers spend a lot of times studying the specific movements and demands of golf so they can do a good job.
I imagine that different instructors would react to this suggestion in different ways. Some might think that consulting another person on class design would violate the hierarchical nature of their art. Some might welcome the chance to stay away from certain kinds of decision making, especially if their confidence in their ability to coach strength development is relatively low.
If you are a student of martial arts you may be in a position of being your own strength coach. For example, if you have organized classes just once or twice a week, then you're in charge of your own conditioning program. If you're in class six days a week, then that's going to be the bulk of your physical activity.
If you are your own strength coach you have two choices: do it yourself or find someone to help you. If you want to get good results you will need good guidance. That means either learning a lot of information about strength or finding a coach/trainer who can offer you quality programming. Either way, make sure that your training is both efficient and specific to your needs and you'll be able to make the best progress in your chosen art.