"Tivo doesn't stay on standby but turns itself back on if left." - does this happen overnight, or at other times?
"Crashes often, showing no picture. This is seen to happen when I turn on the tv after it has been off for a time." - possibly a fault, but could you explain more?
"Some recorded programs skip and freeze" - does this also happen with paused & delayed TV? Could be hard drive issues if so.
"Sometimes live tv picture skips and judders." - pixellation & picture breakup could be signal issues.
In any of these cases, if a reboot solves them for a short time before they recur then there's clearly an underlying issue needing addressed.
Worrying about a 360 is jumping several steps ahead here, you may just need a fault replacement of your TiVo. The V6 box runs the same software as your TiVo but on much newer hardware and would be an upgrade in every sense - but again, we're opening a can of worms here that doesn't necessarily need opening.
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