Use C059 fonts
Lilypond currently uses TeX Gyre Schola (see #4552 (closed)).
There are a number of problems with the TeX Gyre Schola fonts. Let's go through a little history.
The original URW fonts are a direct clone, essentially, of the Adobe PostScript fonts, and so URW's release of Century Schoolbook (from the 1990s) reflects Adobe's original. Accordingly, it is missing many things which we'd like to have, such as /ffi and /ffl ligatures, many accented glyphs, and so on.
URW later released a 2.0 version of the Core 35 fonts (https://github.com/URWTypeFoundry/Core_35). These fonts are available under the LPPL, which is the same license the TeX Gyre fonts are available under. So it is possible from a licensing perspective to include them in exactly the same way the Gyre fonts have been included.
TeX Gyre expands from the original URW 1.0 font set. The team added many more accented glyphs, old-style figures and small capitals. Wonderful, right? Well, in abstract, yes. But the implementation left much to be desired.
For one, for some reason I cannot fathom, the tittles on /i and /j have been moved and shrunken in the TeX Gyre fonts, which makes them different from every version of Century Schoolbook (including both the commercial version and URW's clone). As a result, not only is there a bit of a jarring difference on the /i and /j (noticeable to anyone who pays close attention to the font), but there is also a strong mismatch with the ligatures (see the point to which the /ffi drops).
The old-style figures (not really accessible in Lilypond anyway) are pretty amateurish as well; the small caps have drawn criticism as well (they're also not used by LP). But that aside, look at the Greek. The Greek is unusable. Not only are no accented characters included, but, well, just take a look at a comparison with C059.
Furthermore, C059, unlike TeX Gyre Schola, provides support for Cyrillic, which is completely absent in TeX Gyre Schola.
C059, as opposed to the old URW 1.0 fonts, includes /ffl and /ffi ligatures, many more accented characters, monotonic Greek (fully usable for all modern Greek) and Cyrillic.
The main benefit of TeX Gyre Termes is really in Vietnamese support. But I think it's best to include the typographically-correct version, which is preferable in almost all other situations, as the default. Termes can be included as a fallback, or for manual use.