While downloaders are patched, screen capture is not. Software like OBS Studio (free) or hardware capture cards can record your screen while you play the private video. The quality loss is minimal (1080p is easily achievable), and there is no way for the website to patch your monitor’s output.
Downloading private videos from Thisvid can be challenging due to frequent security "patches" that block standard browser extensions like Video DownloadHelper thisvid private video downloader patched
A long-standing browser addon for Firefox and Chrome that can capture HLS streams and send them to a download manager like JDownloader. While downloaders are patched, screen capture is not