= Solution
Assume the <charmonium> state annihilates through a virtual <photon>, with <lepton universality> and negligible higher-order corrections. For a <vector meson> of <mass> $M$, the <partial decay width> into a <charged lepton> pair contains both the <relativistic two-body decay> factor $\beta_\ell$ and a vector-current <spin> factor:
$$
\Gamma_{\ell\ell}=C\left(1+\frac{2m_\ell^2}{M^2}\right)\sqrt{1-\frac{4m_\ell^2}{M^2}},\qquad
\beta_\ell=\sqrt{1-4m_\ell^2/M^2}.
$$
<First Measurement of the Branching Fraction of the Decay psi(2S) to tau pairs by the BES Collaboration>[PDF p. 3, Eqs. (1)–(2): vector-current lepton-mass correction]{ref} For example, contracting the vector polarization sum with the <Dirac trace> for the two final <fermions> gives a factor proportional to $M^2+2m_\ell^2$, explaining why retaining only $\beta_\ell$ is insufficient. Using $m_\tau=1.777\ \mathrm{GeV}$, $m_\mu=0.1057\ \mathrm{GeV}$ and $M=3.686\ \mathrm{GeV}$ gives
$$
\frac{\Gamma_{\tau\tau}}{\Gamma_{\mu\mu}}\simeq0.388,\qquad
\boxed{\mathcal B_{\tau\tau}\simeq0.008\times0.388=3.10\times10^{-3}=0.31\%.}
$$
The supplied total <decay width> also gives $\Gamma_{\mu\mu}=2.352\ \mathrm{keV}$ and $\Gamma_{\tau\tau}\simeq0.91\ \mathrm{keV}$. It cancels from the <branching fraction> ratio.
The decisive change at $3.770\ \mathrm{GeV}$ is crossing the open-charm <D meson>-pair threshold, approximately $3.73\ \mathrm{GeV}$. The <strong interaction> can now create a light <quark>-<antiquark> pair and rearrange $c\bar c$ into $D\bar D$, without annihilating the heavy pair. This increases the total <decay width> from hundreds of keV to tens of MeV. The <leptonic decay> <partial decay width> is additionally smaller because $\psi(3770)$ is predominantly a $1\,{}^3D_1$ state, whereas $\psi(3686)$ is predominantly $2\,{}^3S_1$; the short-distance annihilation overlap is suppressed for nonzero <orbital angular momentum>. A roughly hundredfold increase in total <decay width> together with an order-ten reduction in leptonic <partial decay width> explains the order-thousand reduction in <branching fraction>. Attributing all three orders of magnitude to the threshold alone would miss the second effect.<Review of Particle Physics by Particle Data Group (2022)>[Charmonium listings: total and leptonic widths of psi(2S) and psi(3770)]{ref}
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