The video seems to be some sort of sync issue, rather than a hardware issue (such as a failed HDMI output) since it happens with a button press (and a reboot fixes it). So to troubleshoot, let's try the following:
The first thing I would do is disable the HDMI CEC feature (Remote Control Setup > Wake with TiVo Button = No). Then use another remote (like the TV remote to control the TV -- or the TV power and volume buttons on the Tivo Remote, if you've programmed that). As I said above, the HDMI CEC has caused issues for many people so let's just try without it for a while...
The second thing I would do is set the TiVo to always output the native resolution of your TV (likely either 4k or 1080p/60fps) and DO NOT use AUTO. Auto has no benefit for flat panel (fixed pixel displays). In addition, for now, turn off HDR (just as a precaution). This combination of a fixed resolution and not enabling HDR will prevent the TiVo from switching resolutions and/or color spaces, which could be causing the TV to lose sync.
If doing both of these things solves the problem, then after a few weeks of usage (so you are really sure it's fixed), you can then try to figure out if one of the above can be re-enabled. I'd probably leave the fixed resolution, and unless you have a 4k HDR display, leaving HDR off probably won't matter. Even if you have a 4K HDR display, the only sources that can use that are 4K streaming services, and most content isn't HDR (and you'd probably not notice the difference). Always outputting only the native display resolution is highly recommended as your TiVo probably has a better video processor than your TV does.
Personally, I don't use the HDMI CEC on any of my setups (we have 4 setups in my house), but, you can try to re-enable it, if you leave the fixed resolution and HDR set to off.
If you still have the issue after making both changes, I would try another TV with the TiVo. You don't know for sure if the TiVO really isn't outputting something or the TV is the issue. One of my Panasonic plasmas periodically shows no signal when the system is turned on (via a universal remote that outputs a script of power and input setting commands to my TiVo, TV and AV receiver), and it's always fixable by turning off the AV receiver and TV -- I've never had to reboot the TiVo to fix it.
The remote issue seems to be something independent. All I can suggest there is take the batteries out, press a few random buttons on the remote with no batteries in it, and leave them out for another hour (or overnight). Then insert a FRESH set of NEW batteries. While the Vox remote is BlueTooth (and possibly two-way), I don't know that the TiVo can do anything to change the remote behavior from BT to IR. So I REALLY DOUBT that the TiVo is causing the remote issue -- rather I think the remote may have its own issues (that hopefully a discharge of any stored energy and a fresh set of batteries will fix). If that doesn't fix the problem, the only other option may be to try another remote...