Is this a good idea? Or am I needlessly wearing out 40 year old irreplaceable parts? Do solid-state electronics age quicker when powered, or does that not matter?
Based on you saying that you don't use the device regularly or very often (i.e. at least daily) I'd say it's not a very good idea.
In general, keeping instruments powered on is a good idea if the device is used regularly (i.e. daily), as it limits the thermal stress from the change between hot and cold cycles. However, continuously powered on instruments also consume their life even when idle, which isn't much of a problem with newer kit that is young and at the beginning of it's useful service life, but for old gear that has more or less reached the end of its life keeping things powered up is pretty wasteful.
Keeping your DMM powered on may not draw much power (or an amount that concerns you), but you're wearing out components that by now are probably already very close to the end of their natural lifetime. Capacitors do "wear out", as do other components like transistors and ICs. Some are easily replaceable, others maybe not. With gear that old, there's a good chance that you consume the remaining life of components for basically nothing.
And I didn't even touch the environmental/ethical impact (i.e. climate change and such) of such wasteful behavior (i.e. creating CO2 for essentially no purpose).
For such an old device and considering your usage profile, you're much better off by keeping it powered down until you really need it, and when you need to use the DMM keep it powered up for a few hours before powering it down again.