This self-answer contains the best information I was able to find. The most detailed information I could find on binding energies was for $\text{H}_2$ (Kari 1973), while my only information on bonding versus anti-bonding is for $\text{H}_2^+$ (Schmidtke 1970).
Schmidtke set out to answer almost exactly the question posed here:
In his review on "The Physical Nature of the Chemical Bond" RUEDENBERG concludes that chemical bonding in terms of potential and kinetic energy
change is not correctly understood unless the validity
of the virial theorem is preserved The seemingly
incompatibility of the opinions that bonding is
either due to a decrease of the kinetic energy (HELLMANN 2) or is originated from a drop of potential
energy (see e. g. PITZER 3) could be clarified by a
closer analysis of various energy contributions upon
molecule formation.
Here is his graph of the results of his model for $\text{H}_2^+$:
The horizontal axis is the internuclear distance in units of the Bohr radius, while the vertical axis shows the energies in Hartree units, 1 a.u.=27.2 eV. The notation is b for bonding, a for anti-bonding, T for kinetic energy, and V for potential energy. At a distance equal to the equilibrium distance for the bound molecule, the difference in T is about 0.2 a.u., and the difference in V is also about 0.2 a.u., so in this sense the antibonding effect is about equally due to the contributions of kinetic and potential energies. At slightly larger internuclear distances, the V graphs actually cross one another, so that at these distances, the antibonding effect is almost entirely kinetic.
For binding energies, we're comparing one equilibrium state with another equilibrium state, so these energies become direct physical observables, and the calculations in Kari are probably easier to interpret, although they seem to agree with Schmidtke's results. For the differences between the bound state and the state where the atoms are separated, Kari has:
$\Delta V_\text{nuc}=+0.714$ a.u.
$\Delta V_\text{el}=-1.063$ a.u.
$\Delta T_\text{el}=+0.175$ a.u.
So here it seems to be somewhat more accurate to use the commonly encountered explanation that binding is mainly an effect due to $\Delta V_\text{el}$. However, I don't think it's accurate at all when people explain the bonding-antibonding effect as if it were entirely due to $\Delta V_\text{el}$. I think they do this because it avoids the explicit use of any quantum mechanics via the de Broglie relation or the Schrodinger equation, but that's just wrong.
It also seems that the small value of $\Delta T_\text{el}$ is due to some kind of very delicate cancellation in the case of $\text{H}_2$ and $\text{H}_2^+$. On the one hand, we expect if the pair of nuclei acts sort of like $Z=2$, then all energies should scale up by a factor of four, and to the extent that the virial approximation is valid, this would also mean that all kinetic energies should go up by a factor of 4. On the other hand, the particle-in-a-box argument will tend to reduce the kinetic energy of the bonding state. In the approximation that this is a rectangular box with sides $L\times L\times 2L$, doubling the length of one side should reduce all energies by $2.25/3$. By this crude estimate, the kinetic energy would then increase by a factor of about three. In reality, it stays almost exactly the same, so the two effects seem to almost exactly cancel. I wonder whether this still holds at all in other bonds besides the H-H bond.
References
Kari, "Complete variational treatment of the hydrogen molecule: I. The full electronic Hamiltonian," Can. J. Chem 51 (1973) 2055, https://www.nrcresearchpress.com/doi/pdf/10.1139/v73-306
Schmidtke, "Kinetic and potential energy partitioning for antibonding molecular orbitals," Z. Naturforsch 25a (1970) 542, http://zfn.mpdl.mpg.de/data/Reihe_A/25/ZNA-1970-25a-0542.pdf
