My view is that if you are satisfied with the results achieved from using your bread machine, you shouldn't feel compelled to rush out and buy a stand mixer unless, of course, you foresee sufficient other uses for the stand mixer. Not all bread machines are alike but even the ones that are prone to kneading the dough too long or producing too much heat in the dough (as is the case with my machine), there are usually ways to compensate for these effects, as I discussed in the posts I linked in my earlier reply. Many people, including professionals like Tom Lehmann and Evelyne Slomon, tend to feel that a pizza dough should be slightly underkneaded to produce the best results, including a crumb that is light and airy with a lot of irregularly-sized and shaped "voids". If a dough is kneaded too much, as many bread machines are prone to do, then the crumb can be "tight" or "dense", with small voids of similar size and shape. From what you have said, the crumb you have been getting is quite good. So, you may not need a stand mixer.
From my perspective, one advantage to a stand mixer is that it is somewhat easier to control the sequencing of ingredients and in controlling finished dough temperature. When using the bread machine, I tend to use the sequencing of ingredients specified by the manufacturer. However, I suspect that it is possible to alter that sequence. I just haven't devoted much thought to doing so. In some respects, I think a bread machine with the right kneading paddles (shape and spacing) does a better job of kneading than many stand mixers. However, if one uses a spiral hook with a stand mixer, such as with some of the more recent KitchenAid models, then the mixing and kneading of the dough ingredients should be much improved over those models, like mine, using the C-hook. You might be able to make a larger dough batch with a good stand mixer than with many bread machines, but if you aren't making a lot of pizzas at one time, then that advantage goes away. If you decide to go with a stand mixer, I would take a close look at those models with the spiral hook. Those models will usually be more expensive, but amortized over time the added cost is quite minimal.