Oh sure, we can try it, and it might even be entertaining, but it'll almost certainly be really unbalanced. There's nothing stopping us from running tournaments using FFA or playing Smash Run tournaments or whatever else, it's just that those modes don't have the same depth that the main competitive mode does and thus people don't tend to stay interested in them as long, ultimately resulting in fewer signups.
You can definitely try persuading your TO to run and stream such a tournament (or do it yourself), and if people have fun maybe there's some future in it, but my hopes aren't high.
The issue imo wouldn't be a lack of depth. Truly, equipments add a lot of depth. So much most players will probably never even be able to see the freaking bottom, and to be able to see it they need to jump through a series of hoops that has nothing to do with being competitive. And that's the issue, and why people wouldn't join.
If we assume the conditions were met to create their ideal sets, and they had ample time to practice... It'd be very deep with a plethora of different playstyles, mindsets, and demand for adaption. So there is potential there.
The issue is how we get there. Generally to construct a competitive environment, it's structured so all participants have the same basic resources and potential. This is why women don't play men's sports, and why we have different weight classes in boxing. So we can keep things as balanced as possible.
Randomization, grinding, variances, or combinations thereof are not inherently detrimental to competition. In fact, in cases like pokemon(gen 6) it's good that people can't just think up a dream team and have it as soon as they do. This would dilute the competitive experience, and discourage people actually researching on team ideas and how best to train them.
However, pre gen 6 and in games like Kid Icarus Uprising, this effect was too great. People had to dedicate entire months/lives to the game to develop themselves, and when they did it wasn't like they'd only use that one team/weapon forever. And they also couldn't spend enough time developing their own skills in that time frame.
This equipment system does just that. If it were less randomized, perhaps came with a list of equipment and which characters were more likely to receive them (or even beating those characters in classic mode) where there was some method to the madness, so that it was feasible for someone to pick up the game and be prepared to at least TRAIN with people more experienced in the game within a foreseeable time frame, then we could allow ourselves to be more optomistic.
All that said, there is one way I can foresee equipment becoming tournament legal.
If Datel Powersaves ever releases an equipment unlock code, and TOs provide the Powersaves at registration (takes a minute to overwrite, and it's permanent) there'd be no reason to exclude them in their entirety unless we find the equipments themselves are broken/uncompetitive. Because if the best of the best are available, there's no reason to exclude people who got theirs legitimately too (while possibly weaker)
With fusion, you had actual control over what happens. Now you just have to grind for hours for a chance at said perfect equipment.
You had no control over the drops you got aside from their general values, and the way the stars/mods transfer were also essentially random. The only control you had was whether or not to fuse, and which two weapons to fuse. You had no control over what happened at all. The game showed you what it would make, what mods it would take off and if it took off the wrong ones, oh well.
You could feasibly get a particular weapon, maybe with a mod or two and with decent stars. But overall it'd likely still be extremely inefficient. Just like here.
I enjoyed KI:U endlessly, and made some great weapons under this system (was probably the one of the first to realize eyetracks + shot range + shot homing = GG) but only until i started cloning did I start making the perfect weapons and did the headaches cease.
