How safe is it to supply power to a linear regulator output?
In general it is not safe since most of the time regulator properties are not guaranteed when the input side isn't powered on. Some regulators are specified to be powered 'into the load', as in, with an already present voltage on the output.
The diode trick for the 78xx regulator (which was IIRC actually in the datasheet) is the minimum to keep the part biased but there could be other problems too depending on the internal topology. For example most regulator can only supply power but not sink: this is not universally true, however.
I think it should be decided case by case