If they intend to apply taiji, then yeah I think they do. And if they don't intend to apply taiji, then no it isn't a martial art in that particular case. Just my view. It's entirely possible to practice many martial arts in a way that conveys very little martial prowess. Taiji practiced in a park every morning may carry lots of healt benefits, but will yield very little insight on how to use taiji to defend yourself.
Hmm, gonna disagree with you, there, Stuart. Respectfully, you understand, but, well, you're wrong!
Okay, I'll be a little more serious.
As I've said a number of times, there are as many reasons to train in martial arts as there are martial art students, and not all of them train in them to "apply" them. Not all arts are designed to be "applied" in such a manner, either. Ken's already mentioned Iaido and Jodo, and I'll expand on Bruno's answer in a moment, but to that you can also add Kyudo, pretty much all Koryu systems, and many more. Let's take Kyudo.
Kyudo is trained as a solo experience with the aim of perfect unison between self, the bow, the arrow, and the target, even beyond the idea of accuracy in the shot itself. There is no opponent, there is only the target and yourself. But then again, the art teaches you how to draw a bow, notch the arrow, aim accurately, and fire towards a target. It certainly covers both the description of "martial" and "art", to my mind.
Now, if I understand your argument, you are claiming that if it is removed from combative purpose, or self defence application, then it ceases to be a martial art. I'd actually argue the opposite. And it comes down to the application of terminology.
For me, a martial art is beyond simple technically applicable concepts, frankly it has outgrown such base ideologies. Military skills and methods are not martial arts per se, they are military skills and methods. Self defence skills are devoid of the breadth of conceptual detail and knowledge, the depth of subtlty that martial arts encapsulate. They are by necessity simple, gross motor, reliable. Maybe if we think of them as cars (sorry, watching Top Gear right now....), that may help. A military approach can be either a tank, or a jeep. A jeep can have some civilian usage, with soem sacrifice and a fair amount of compromise, but a tank really doesn't. Self defence is a basic, sturdy vehicle, maybe a Ford Transit van for instance. Martial arts, on the other hand, are the luxury and high-end machines, filled with things that are not really necessary, move away from the pure practicalities of the van, in order to give various experiences. And there are as many of these luxury vehicles as there are drivers, some customised to the specific needs of the driver, others more considered "classics", others factory-standard but well-equipped. Hmm, may have gone off on a tangent there....
So it really does come down to why someone is training in a particular art themselves. If it is for self defence, then that should be the priority. If it is sport, then that should be the priority.
Now, as it comes to sparring, I think you and I have been here before, but here we go again! For me, sparring is far from ideal, as it is completely removed from the reality of what I train for, in all the myriad forms that I do. My reasons are many, and listed in various other threads here, but in brief, my biggest complaint is the lack of reality involved. That is closely followed by the restrictions on applicable tactics and expressions of the strategic methods of my art.
This is different to saying that free-form, spontaneous expression of tactics and technique against random, unnominated attacks, all the way to full speed and power, pressure testing, and so forth are bad. In fact, I think they're essential, especially if you're looking at defensive skills being developed. But the common cry of "it's as close as you can get" is frankly wrong. It's not really close at all, when you get down to it. And there are better methods that are far closer to reality and the skill development required. Once again, these are detailed on other threads, but if you ask, I'll go through them here again.
In terms of the tactics that Bruno was talking about, sparring demands certain things, like staying involved in the situation when escape is possible or advisable (and it's one of our first choices, really!). To truly apply the tactics and strategies of our art, frankly, you should be able to run away! And that isn't really given as an option in sparring.
So, uh, no. Sparring is not essential. Unless, of course, it is preparation for sparring-style application (such as sporting competition). But testing of applicability and skills under pressure in a variety of free-form scenario drills, if such applications are your aim, absolutely is.