Solar thermal

Solar to heat is very efficient, but then heat to electricity is problematic, and even in a very good system, the efficiency is likely to be low unless you're able to create a very large thermal difference, in which case you're talking about a concentrator system, and this will have the effect of losing access to diffuse radiation and making the system limited by direct normal radiation, which you cite as a weakness for PV. It's only a weakness for CPV. You can't get an efficient heat engine running off a 100C or so temperature difference, so you need to have solar concentration to achieve any sort of good heat engine efficiency for electrical production. Also, PV does use quite a bit of the spectrum. I think you may be confusing the band gap energy with the absorption spectrum. Photons with energies above the band gap will also be absorbed, but the additional energy beyond that of the band gap energy will be quickly lost due to carrier thermalization. Still, an Si cell with a 1.1 eV band gap will absorb and produce power out of a, for example, a 450 nm photon with an energy of eV, it's just that the carriers will relax to the band margins very rapidly, and so the remaining carrier energy separation after relaxation is 1.1 eV. In other words, efficiency of conversion drops off the more you go above the band gap energy, but absorption and conversion to electricity still occurs. Multi-junction cells, such as those used in CPV can achieve much better efficiency across the spectrum since shorter wavelengths with more energy are absorbed by a higher band gap material, and less power is lost in carrier thermalization. A single junction will still use light from across the spectrum, as long as it's above the band gap energy, but just not as efficiently. So, an Si cell will actually handle quite a bit of the near infrared, out to around 1100 nm in wavelength. There is some power left in the longer wavelengths, but not that much. If you look at the Shockley-Quessier limit, Si is actually quite well positioned as far as band gap to obtain maximum single junction power conversion. If a successful hot carrier extraction method can be developed, one which also allows for good absorption, a single junction Si cell could be very, very efficient across the spectrum, depending on how quickly hot carriers can be transported.

So, I do think that, especially for small scale applications, if you look at the electrical output for PV versus the electrical output for a heat engine idea, PV has very clear advantages. If you want to talk about concentration of solar power for high delta-T heat engines, then you can get performance on par with that of CPV, and possibly at a lower cost per kWe with further development of the manufacturing of solar heat engines, but then, just like with CPV, you are dependent on the direct normal radiation and the quality of your tracking and concentrating.

For domestic hot water, heating, and industrial hot process water, solar thermal is of course much, much more efficient than, for example PV and resistance heating, but if you're talking to electrical output, untracked fixed panel PV has the upper hand over untracked fixed panel solar thermal and use of a low delta-T heat engine.