if you don't have any boolean useful (that you want to keep) fields, you could use the autofield+dynamic select.
A field/column with all null rows is detected by the autofield tool as boolean (the autofield tool assigns the smallest size possible, that means boolean). After that you can use a dynamic select to select everything but not boolean fields.
@mjunior7 I use @ned_blog's macro from this (old) post. I haven't tried them personally before today, but it looks like @AlexK has one (similar to Ned's) and @Hollingsworth has another one (he's added some cool functionality) on the gallery.
@JordanB Thanks! This workflow helped a great deal. I have some blank cells instead of null cells so using CountNonBlank in the Summarize Tool helped capture both blanks and nulls. If you also wanted to remove any columns that had zeros, in addition to blanks and nulls, I'm not sure how to apply that to your approach. But otherwise, very helpful!