Okay, here's what I notice:
1. You have two copies of GregorioTeX installed: the one which comes as
part of TeXLive (in texmf-dist) and one which you installed manually (in
texmf-local). Both are at 5.1.1, so this is redundant.
2. The log from your document indicates that the texmf-dist version is
the one being loaded. This is out of order. The texmf-local version
should have priority based on the order in TEXINPUTS.lualatex. This
suggests that the ls-R file for texmf-local is out of date (the !! in
front of the texmf-local path indicates that TeX uses an ls-R index for
that folder rather than searching it outright).
3. The executable for the texmf-dist version is not showing up,
suggesting that it's not on PATH. Since the executable should be
matched to version, it maybe that the "can't find" message is caused by
this. However, since the texmf-local version of the executable does
show up, and it's an identical version, I wouldn't expect this to be a
problem.
So, what to do? Well, there are several options (presented in the order
of my preference):
1) You could try to get LuaTeX to find the texmf-local installation
first by running texhash. This is supposed to be run by the installer,
but clearly something went wrong in this process. Perhaps a permission
error? You might try running it as an administrator if running it as a
user doesn't fix the problem.
2) Since the texmf-local install is redundant, you could uninstall it
and make use of the texmf-dist version. My only concern here is the
fact that the executable for this version doesn't show on PATH. Without
knowing why this is, I don't know for sure if simply uninstalling the
texmf-local version fix this. If this file was removed by the
installer, then the uninstall process should restore this file, but if
there is some other cause, then you could be left without the executable
altogether. To fix that you would need to make use of tlmgr and I'm not
sure what the correct incantation for that would be off the top of my head.
3) Speaking of tlmgr, you could try repairing the texmf-dist version
without touching the texmf-local one. I don't like this option because
it leaves the texmf-local version in place to cause potential problems
down the road. Besides, as I indicated before, I'm not sure of what the
correct incantation is.