I agree with @skoestlmeier's answer that the $\sqrt{252}$ should be considered dimensionless. However, using the result from your dimensional analysis (i.e. returns and volatility having dimension $time^{-1}$), you would conclude that the black scholes (BS) formula is dimensionally inconsistent (see this question).
I find it more helpful to think of returns as dimensionless (i.e. ratios) and therefore volatility, given a measurement frequency. A naive analogy would be, if you have a car the travels at constant velocity $10\,m/s$, then it will have travelled $10\,m$ after $1\,s$. The $10\,m$ distance is the way of thinking about returns, given you always use some frequency (1s, 1m, 1y etc..).
Now if you must give them dimensions, then I suggest thinking the notions of returns and variance having a "unit" $time^{-1}$ (much like the velocity in my naive example) since they are usually quoted in annual terms (which is a choice of frequency). As a consequence volatility must have dimension $time^{-1/2}$ which is in agreement with your stochastic calculus source and you have a dimensionally correct BS formula.
(P.S. I do recognise the inconsistency there would be when plugging in the variance formula, so I welcome any better intuitions.)