Okay. So, final update on this.
Update: New Problem - Temporary Files like Theme Custom Logos are Being Created With Web as Owner.
I have not been able to fix the file creation issue. Instead, I just brute forced it through the simplest solution I could think of. I just turned my logo into an SVG file, renamed it as the default, and replaced it in the themes/quark/images folder.
As a reminder, the problem was that the admin panel option to upload a custom logo did not work in NFS because the process involved creation of a temporary file with web as owner and group. This led to Grav erroring because it did not have permission to further manipulate the temporary file.
I ended up just brute forcing this the same way as replacing the favicon instead of attempting to redirect.
Besides from this issue, Grav otherwise seems to work. I have been able to create pages and sub-pages for blogging. The admin panel works well for these processes.
Conclusion: With some minor workarounds, it is workable to run Grav on NearlyFreeSpeech. If one follows the above steps, you should be up and running without issue.