My wife had our son go through this elaborate process of mixing the
ash with specific proportions of regular dirt (mostly clay in our
not-so-arable area) and sand, saying that the ashes can provide useful
nutrients to the garden but in high concentrations can be destructive.

That's a good rule of thumb for a lot of things you are tempted to add
to a garden.

The more "scientific" way would actually involve measuring what your
particular ash is like, what your soil is like, etc. Not necessarily
a bad idea. But at some point one usually ends up with a certain
amount of guesswork and "well, things seemed to grow when we did it".