Yes, it seems it really did the trick. Followed this guide and transfer rate in the file manager in Kodi’s system menu increased from 8.7 MB/s to 11.7 MB/s. Movies that were stuttering now are playing just fine - at least the first minutes, where previously they would have hanged for at least 5 times. Just one movie hanged, but it was kinda expected, since it has a medium 86 Mbps bitrate. If I want to play that, I’ll need a local copy.
Now up to change the locations of my library’s folders, and mount the folder on my other Kodi box. Just hope there will be no problem with the MySQL database.