I managed to solve this on my own.
By chance, amidst some thought process, I ran df, the Linux command to display disk fragmentation and space allocation. I noticed the disk partition that contained /tmp was at 98%, then going back to 1%, then approaching 98%, and doing this nonstop.
Suspiciously, the moment of this transition (somewhat) coincided with the “No scap dp present” in the log. I’m unsure what the original size of the partition was, but, after increasing it by 5 GB, the errors in the log stopped. A few minutes afterwards, GSA (the web interface) had been rid of the red flag “The SCAP database is required” and displayed NVTs correctly.