some people claim their Tivo's run better after deleting lots of content - maybe that frees up big blocks of empty space, but i see you already tried that.
I'd ask for a new box & binge watch what you have recorded, while that order is progressing.. Also disable "record recommendations" to save on disc wear and tear + keep it on standby when not watching stuff
it is more likely that the kids never use standby so that their hard drive is recording stuff 24/7 , as it always buffers the currently tuned channel plus fills itself with recorded suggestions. that has to wear out the drive in time, even if it has some smart auto-defrag coding/.
and if records current channel AND a suggestion simultaneously on an almost full drive, that could mean lots of hard seeks ? = bad design IMHO
there are some "magic" key press combos that clear local cache if you can find the threads & try them...
I've run some diagnostics on your equipment and I can see an issue with the hard drive on one of your boxes that is likely causing the issue. I've sent you a private message (purple envelope at the top right of your screen) so I can arrange for a technician to take a look.
Tech fan? Have you read our Digital life blog yet? Check it out