Have you tried your tester on any other CRTs? That might tell you how much weight to give the tester's opinion, especially if you test some known-good CRTs as well as a dud or two. Testers vary in quality/sophistication, like other devices.
In my experience, CRT testers don't always tell you the whole story. I have seen CRTs that look weak on a tester, but make a fine picture when used in a TV. The best tester of all is a television.
Also, if a CRT hasn't been used in a long time, it's common for it to "wake up" and show stronger emission after it has cooked for a while on a tester at normal or slightly elevated filament voltage.
Phil Nelson
Phil's Old Radios
https://antiqueradio.org/index.html