Regarding advanced searching: is there a way to add a custom field to search results regardless of the value? Right now the closest workaround is to “search” for “With any value” on the custom field, but this filters out tasks with an unset custom field.

Yes. Use “Is” without filling the other field =) I believe that does what you need.
Wow. Thank you, @Bastien_Siebman, for shining light on that! Does exactly what’s needed.
So this (“Machine” is a custom field):

means “show the Machine column with all or no values”.