As far as mobility related armour sets go, I'd much rather have the new shadow armour than a hypothetical ninja set with a dash, even in classic. The acceleration boost (that even works in midair) is imo a far more significant bonus than a dash, and the +10 defence on shadow armour compared to ninja is significant too. I'm mainly saying this to refute the point that ninja armour with a dash would "break" classic, since we already have a better mobility related armour set that doesn't really break much at all (sure it comes a bit later in the game, but dashes aren't too useful on classic mode EoC and evil bosses).
Attempting to dodge the bosses in Classic Mode is actually the
weakest way to exploit them, as running in a single direction (the bosses are very slow, and very stupid), stacking defense to become unkillable (they deal hardly any damage), or bursting them down with damage before they can become a threat (they have fractions of the HP they do in higher difficulties) are all vastly more effective ways of dealing with bosses in classic mode than trying to engage with them and avoid their attacks. Classic having more mobility tools will not change much, I don't think. Also, like you mentioned, Shield of Cthulhu has the bash feature, so I don't think Ninja Armor having a simple dash will be outclassing anything.
Honestly, I like the idea of having a dash in classic mode, as the strategies I just mentioned fall off hard (still possible, just more difficult) in higher difficulties, while learning how to actually dodge the bosses becomes
more valuable; giving a dash to Ninja Armor would be a great encouragement to experiment with mobility and dodging so Expert Mode isn't as much of a shock;
My first playthrough 10 years ago on Mobile saw me equipping the highest-defense armor, swinging my sword, and repeatedly flinging myself at enemies until one of us died; this strategy almost always resulted in my victory, even against bosses, and the game was generally quite easy for me. Expert Mode doesn't tolerate such buffoonery, and as a result, my first playthrough on the mode was quite a shock, forcing me to learn how to engage with the enemies and bosses rather than running right through them like I did in classic. Allowing players in Classic Mode to get the hang of the dodging methods they will likely be using in Expert Mode seems like a pretty good idea to me.