Your MMA class is Boxing, footwork and Muay Thai not MMA. It's what we call a 'stand up' class.
Agreed. If you have no grappling/groundwork mixed in, I wouldn't consider it MMA. I should say that I view MMA as primarily a sport method--not primarily a self-defense method.
MMA is more than just boxing, MT and BJJ though, for it to be MMA it also needs elements of other martial arts, karate, Judo, Aikido etc etc. The point of having MMA is to have as much ammunition as you can by knowing as much as you can and putting it together.
I don't see this the same way though. I see MMA as a spare, stripped-down system drawn from parts of much larger systems and ultimately containing relatively few techniques but combined in a way that works for you. It's not about having as much ammunition as you can by knowing as much as you can but about having the right ammunition for you by drawing from as much as you need. In practice people do very well with boxing, Muay Thai, BJJ, and wrestling. What you're describing I would characterize as more of a JKD approach--but even then I expect it to be ultimately reduced, though not as much as in MMA.