If anybody else reaches this place, especially for the Remote VSCode server enthusiasts, I found out how stupid I was and why Intellisense was not working at all.

I checked the .cache folder. Because the remote machine was very low on disk space for the home directories, I created a symlink to a local SSD for a similar .cache folder. The problem was, I deleted the .cache folder from the SSD (for refreshing the Intellisense database that no longer worked properly) and the symlink remained dangling and could not create any new data inside.

Took me no more then 3 months to figure that out. So yeah, check for file descriptor access limits, symlink, or simply disk space where the cache is stored.

Kept only the C/C++ Extension Pack extension. Having also C/C++ extension seemed to conflict with the other one so for now, seems clean.