I like the previous suggestion of Capoeira, because it does emphasize physical fitness, does not involve contact during sparring, and can be practiced alone. But as I said in one of the comments, I think it can be pretty intimidating for a beginner to begin taking Capoeira classes, because it does involve a lot of techniques that seem advanced. To someone coming in with poor physical fitness and someone who is a bit older, the emphasis on acrobatic / gymnastic skill may seem a bit off-putting. Worth checking out, though, if you can find it near you.
Kickboxing wouldn't work. It involves punches and kicks to the head, which the original poster said he didn't want.
Cardio kickboxing may work, though. In cardio kickboxing, the classes are all about getting your heart rate up while performing kickboxing techniques. It's aerobics but with martial arts techniques. There's no contact. And there's usually no pretense about it being good for practical self-defense. You can find cardio kickboxing classes taught in all kinds of martial arts schools and boxing/kickboxing/mma gyms as a separate class. Many Krav Maga schools also have a cardio kickboxing class. And it might be called something other than "cardio kickboxing". So you do have to look around, go to the web sites of different schools near you and see what they offer.
There's also regular boxing classes, but with no contact. Many boxing gyms do offer a no-contact version of boxing training. It might be called "boxing basics" or "boxing conditioning". They may even have a requirement that you take this class to begin with for a few months before moving on to sparring classes. But a lot of people just do the conditioning classes and don't bother with sparring at all. A good boxing coach can tailor your conditioning to your body type and where you're at in your current conditioning level. You can also pay for private boxing coaching however many times a week, so you get the coach's undivided attention.
Some of the things they'll have you do in boxing conditioning classes or private coaching include: jumping rope, learning the basic punches and stances, learning combinations, practicing punches and combinations on the bag and speed bag, practicing with focus mitts, non-contact or very light sparring, weight training, the weighted sled, tire rolling, the battle rope, running, push-ups, sit-ups, pull-ups, etc. You'll start getting a fighter's physique after a year of doing this 3 times a week.
Another possibility I think might work is Contemporary Wushu. Wushu is a sport that involves repeating very intricate and difficult martial arts choreography (forms / taolu). There's no contact and no sparring. The emphasis is on performance, not on self-defense. And the forms are generally practiced alone. The physical requirements are higher as you get more advanced, so there's a gradual progression. And there aren't many fat people you'll see in Contemporary Wushu, because it does cultivate physical fitness. This satisfies all of the original poster's requirements. And having already done Taiji, this may feel familiar.
Finding a competent Wushu instructor may be difficult, depending on where you live. But if you can get to one, it might be worth checking out.
You can read up about and watch videos on Wushu at:
Be careful, though. Contemporary Wushu (usually called "Wushu" in the West) is not the same as Traditional Wushu (which in the West is usually called "kung-fu"). Traditional Wushu also involves forms, but often the emphasis is less on performance and more on self-defense. Since the original poster did not care for self-defense, I recommend sticking with Contemporary Wushu instead of Traditional.
Those are my top 3 picks: Cardio kickboxing, boxing conditioning, or Contemporary Wushu.
Hope that helps.