Which is most reliable excitation energy: LUMO-HOMO energy or TD-DFT excitation energies?
When you can, use TDDFT, in part because of the oscillator strengths

I like to point out to my group that optical excitations consist of two things consistently (e.g., in a UV/Vis spectra). There are the excitation energies (the x-axis) and the intensity or extinction coefficient (the y-axis). To some degree, the peak widths give a bit more information (e.g., into vibrational structure).

The example you give above is a great example. Most of the excitations are forbidden and have no oscillator strength. I'd pick the same excitation, although I'm curious what the molecule is - usually there are much higher oscillator strengths.

You're right that there are papers and groups that use the HOMO and LUMO eigenvalues. While some DFT methods (e.g. optimally tuned range-separated hybrids) do provide HOMO-LUMO energies that correspond nicely with experiments, there's nothing physical about Kohn-Sham orbital eigenvalues. Moreover, in general, unfilled orbitals are not handled well by DFT methods.

It's been a while since I looked for reviews, but general TDDFT excitation energies are ~0.2-0.3 eV from experimental absorption energies after correcting for solvent effects, etc.

In my group, we also do what I call a "consensus" model. That is, when you run a TDDFT calculation, you get the HOMO and LUMO energies anyway. Since there are slightly different random and systematic errors from each approach, you can calibrate both against experiment and use a multivariate linear regression. The result decreases the average error significantly.