Shut down or leave on?
**snip**
I hoped to reduce the probability of it [tape player]
failing at the same time as the internal HD by
starting it less often
For HDs the R/W head always comes into contact with the disc once the rotation drops enough. Crack open a dead HD and you'll see nice concentric rings which just don't exist on a new device. HD life depends an awful lot on the bearing condition as well; I found older bearings (HDs 10 years ago) get a bit gummy and may stick on startup, especially in humid conditions, in which case it was better to keep running. Environment is a huge factor - if you get a "condensible atmosphere" as you would frequently get in Hawaii, it's best to keep running because the elevated temperature discourages condensation.
For a tape device the usual problems are mechanical wear (which doesn't happen if you're not running a tape), expired pinch rollers, spiders, geckos, ants, dust, and lint. The dust and lint is usually made worse by a fan forcing dirty air from the room through the drive. The electronics themselves are not likely to fail unless you have a lightning strike. Some electronic components age poorly and there's no saving them regardless of whether you remove power or not.
I've found most HDs manufactured in the past 5 years are very tolerant of power cycling so I think the primary consideration for switching off your equipment in this case would be environmental factors (does it get very humid where you are).
But I notice that most backup utilities are designed to
backup automatically at the same time every day or week.
That seems to assume that the system is always running.
You can initiate a backup yourself; you don't have to have the system running all the time. Most backup software was originally intended for businesses so it's convenient to do the backup when hardly anyone is using the machines; home computers simply aren't used (as much) in the same way.
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