As you've discovered, one can only add up to 16 distinct columns to a Tableau crosstab.
Your best bet is to include some sort of text in your viz / dashboard which tells users they can open up our browse dialog (via "View Data") and let them browse / sort there.
Unless you create some sort of web-page based datagrid which shows the data, and then embed THAT web page in a Tableau dashboard, you won't be able to accomplish this (include a "browse data" dialog appear as a sheet in the workbook).
Unfortunately that does not achieve what I am after. If I instruct users to click on the 'View Data' they cannot filter.
Is there any other way to achieve this in Tableau?