The most likely explanation is that the analyser outputs a non-interlaced signal which is not strictly any of 525/60 or 625/50 standards but close enough for "analogue displays". The AV2HDMI converter is expecting more rigid adherence to one of those.
This is almost certainly the case. These converters are way more fussy than an analog CRT. NTSC is interlaced with quite precise timings, it is very common for computers and video games and such to produce a non-interlaced signal that is close enough to NTSC timings for a CRT to work fine, but it won't work with something that is expecting interlaced NTSC video.
I have used one of those cheap little backup camera LCD monitors for testing early arcade boards and FPGA implementations of them so some LCD monitors do handle these odd signals just fine.
Yes, analog monitors/TVs normally run the horizontal & vertical oscillators constantly, whether there are syncs available or not.
Their free running rate is close to the nominal locked frequencies, & when they receive syncs, being basically PLLs, will lock to them.
Except for very early systems like the old UK 405 line system, the line frequency of analog TV standards are very close & will lock without difficulty.
Field rates may require a "tweak" of the vertical hold preset pot, although the last generation of TVs could recognise the 50Hz & 60 Hz systems & self-adjust.
If a converter requires very precise timing, it might well become "confused".
I am still suspicious that the Field & Line sync signals in the R3265A may not be locked.
This can be easily checked by displaying the video waveform at field rate.
If the H syncs are not locked to the V ones, they will drift through the display, just as they do on the monitor.