I'd like to welcome you to the forums and say I'm really sorry to hear of the trouble you're having with your TiVo®. It really shouldn't be needing to be powered off and back on just in order to get it to work.
From your initial post, it does sound like your TiVo® could be going into power saving mode, if you want you can disable this feature by following these key strokes on your remote control:
Home > Help & Settings > Settings > Devices > Power Saving Settings
Here you can select different settings for power saving options.
I was concerned that you stated that the box would only come back on if you powered it off and back on though, normally after power saving, you would only need to press the standby button on your TiVo® remote.
I did run some testing on your TiVo® and saw that one of the signal strengths were a little high, whilst this shouldn't cause this problem for you, signal strength issues can manifest in different ways.
If my above help with Power saving settings makes no difference and you'd like for me to arrange a technician to call out to check out the signal strength then let me know and I can get that looked into for you.