Could you share your workbook in .tbwx format. I think "re-shaping" your data-set is the way out.
You need to pivot your data.
To get what you want, the data has to be in a single column for Tableau to use it as a filter. Right now, each value you want to filter on is its own column.
This is a common issue with "survey" data. There's an entire group dedicated to that subject, by the way: Survey Data