I know that one of the standard measures for the "tailedness" of a distribution is kurtosis, i.e. fourth standardized central moment $\frac{\mu_4}{\sigma^4}$. This measure is sort of intuitive to me: as the 4th power "punishes" the outliers much more than 2nd power of the variance does, we would expect a distribution with a heavy tail to have a large kurtosis.
Recently, I thought of using another measure of tailedness--computing the probability content "far away" from the mean, let's call this measure $T$ (for "tail"). Then $$T(X) = \mathbb{P}(\lvert X - \mu\rvert\geq\sigma)$$ where $X$ is a random variable whose tailedness we are measuring, with $\mu = \operatorname{E}X$, and $\sigma^2 = \operatorname{Var}(X)$
We could also perhaps use different multiples of $\sigma$ in the definition (which would, however, also bound the value of $T$ to less than $1$ because of Chebyshev's inequality).
Has this measure been used for judging the tailedness of a distribution? What are the properties causing it not to be in (widespread) use? It seems to me like it would be a good way to capture the "Amazon business model phenomenon," where they can exploit the high overall probability content among niche titles: