There is most definitely a direct relationship connection between kurtosis and the tails of distribution. One simple connection is as follows: Let $X$ be any random variable with finite fourth moment, let $Z = (X - \mu)/\sigma$, and let $U = Z^4$. Then the kurtosis of $X$ is $E(U)$.
A common way of teaching and understanding expectation is using the "point of balance" concept, where the expected value of a random variable is equal to the point on the horizontal axis where the distribution plot balances.
Consider the plot of the probability distribution function (pdf) of $U$, and locate the value 3.0 on the horizontal axis. This is the kurtosis of the normal distribution. If the pdf is too heavy to the right of 3.0, so that it is pulled down to the right when attempted to balance at 3.0, then the kurtosis is greater than 3.0, and the distribution of $X$ can be said to be "heavier tailed than the normal distribution." Conversely, if the pdf of $U$ is too light to the right of 3.0, so that the curve is pulled down to the left when attempted to balance at 3.0, then the kurtosis is less than 3.0, and the distribution of $X$ can be said to be "lighter tailed than the normal distribution."
In light of the above discussion, "tail leverage" is perhaps a better characterization of kurtosis than "tail mass." It is possible that more mass in the tails corresponds to less kurtosis, and that less mass in the tails corresponds to greater kurtosis. It all depends on the remoteness of the mass; i.e., it depends on the leverage.
Precise mathematical logic showing that kurtosis is mostly determined by the tails is given as follows:
For all possible distributions of random variables $X$ having finite fourth moment (discrete, continuous, skewed, empirical), kurtosis is necessarily between
$\int Z^4 I(|Z| > 1) dF(Z)$
$\int Z^4 I(|Z| > 1) dF(Z) + 1$,
where $Z = (X-\mu)/\sigma$.
For continuous distributions in this class with density of $Z^2$ decreasing on the interval $[0,1]$, the "$+1$" is reduced to "$+.5$" or less.
Finally, for any sequence of distributions where the kurtosis $k$ tends to infinity,
$\int Z^4 I(|Z| > b) dF(Z) / k \rightarrow 1$,
for all real $b$.
These are all mathematical theorems so there can be no counterexamples.
Thus, it is primarily the values in the tails of the $Z$ distribution that determine the value of kurtosis.
Yes, there are other tailweight measures that are not kurtosis, in fact there are infinitely many. So it is a red herring to state that kurtosis does not measure tailweight if it does not correspond to a particular tailweight measure. It is an unsupported generalization to say that, just because kurtosis does not measure one of infinitely many tailweight measures, that kurtosis itself does not measure tailweight.
I have seen no theorems generally linking kurtosis to the peak, or to the probability concentration within the $\mu \pm \sigma$ range, but I have seen plenty of counterexamples to such notions. Can anyone point to such a theorem?
The three theorems I mentioned above are proven in Westfall, P.H. (2014). Kurtosis as Peakedness, 1905 – 2014. R.I.P. The American Statistician, 68, 191–195.
Edit, 7/18/2018: Yet another theorem to connect kurtosis to the tails (and not the peak). Take any distribution (or data set). Remove the data or mass within one standard deviation of the mean. Replace the data/mass in any way you like, but keep the mean and standard deviation the same as with the original data/distribution. Then the difference between the maximum and minimum kurtosis for all such replacements is $\le$ 0.25.
If you play the same game with the tails, the difference is unbounded.