It depends what you mean by 'traditional.' In the strict sense of the word, boxing, wrestling and Muay Thai are probably the most traditional martial arts. But they are not included in discussions of TMA. Why is that? Because the phrase TMA actually means something other than being old or having a strong culteral heritage. It is generally used to denote eastern martial arts as popularly practiced in the west in the 70s through the 1990s. Most of these styles (including Shotokan and TKD) are actually not very old -- dating back to around the 1930s.
If you took a survey of schools promoting themselves as traditional martial arts, I am very confidant you would see more light/no-contact un-padded sparring than in schools that do not promote themselves as traditional. Do you agree?
Again, what does 'real "traditional'" mean? I would argue that Kano's changes to Jujutsu to create Judo (from which BJJ was derived) was one of the most significant moves away from (what are commonly referred to as) TMA. Judo was a proven effective combat sport & martial art from the start, I wouldn't include it in the list of TMAs that needed vindication after failure in NHB/MMA/Vale Tudo. It always did moderately well.
Indeed, Judo is older than most karate styles including Shotokan. But age isn't generally considered when determining whether an art is traditional. Boxing & wrestling are as old as sports get.
And this was Kano's genius. By training in a way that you can go all-out without causing injuries, your actual fighting ability increases dramatically compared with training dangerous techniques without full resistance & competition. This is in my opinion, the key difference between TMAs (JJJ, most karate, Wing Chun, Wushu, etc.) and non-TMAs (Wrestling, Judo, BJJ, submission grappling, Muay Thai, Sanda, etc.)
Personally, I don't think they are a very good exercise. Better than nothing, but rife with problems compared with other training methods. I would bet on a fighter with 1 year of wrestling over one with 10 years of kata practice any day.