Not really. That analogy assumes they cannot cut without the shinken their instructor recommends (not provides - a different thing). Sword arts are apparently pretty specific and detailed in their differences. I would assume a different shinken - even one that is of good quality - might be ill-suited to a particular style for some reasons. I'd liken it to firearm training. If the instructor is teaching how to effectively fire a lever-action rifle, and you have a bolt-action rifle, you're in the wrong class (or just brought the wrong tool). Again, since the tradition, itself, is part of the reason people train these styles, continuing that tradition seems an important factor.
If someone wanted to teach a generic Japanese swordsmanship class, just focused on drawing the blade and cutting things, that would be a different matter. They'd still need to make sure they were using a quality tool that won't fall apart and has a reasonably proper balance, etc., but they would have a wider range to choose from. Heck, they might even find some modernizations that would fit well with that idea, and might be interested in evolving both the weapon and the usage. I'm not sure why anyone would be much interested in that, though, since it wouldn't actually be more practical than the traditional styles, since none of us are allowed to carry those swords, anyway.