Using INDEX, MATCH, and COUNTIFS to dissect survey results

I issued a survey to communities in my region. I have their results back, but I'm having trouble creating a table that will be easy for others to use to get results they can use.
Right now I've got a long COUNTIFS statement which delineates the three or four criteria I need to search by, but I have so many criteria that change so often that I need to find a way to do it more effectively. Here's what I'm using:

=COUNTIFS('Raw Data'!$BM3:$BM1038,C$2,'Raw Data'!$BL3:$BL1038,$B12)

'Raw Data'!$BM3:$BM1038, points to the column with age information.
C$2, points to the age criteria.
'Raw Data'!$BL3:$BL1038, points to the column with gender information.
$B12, points to the gender criteria.

What I would like to do is have an INDEX and MATCH function that will search the header column of my raw data table for the column name in C$1, "What is your gender?", and then uses the identified column to search for the gender criteria in $B12. I've tried a number of potential solutions with no success so far.

I think it should be something like

=COUNTIFS(INDEX('Raw Data'!$B$2:$CT$1037,0,MATCH(A12,'Raw Data'!$A$1:$CT$1,0)),B12)

And then duplicating the process for the different criteria. Thoughts? Am I on the right track? Can this even work?

Please attach a file (not image) so that respondents (a) do not have to type in data and (b) have data with which test any formulas. Ensure the data is representative of your actual "real world" data.

Make sure there is just enough data to demonstrate your need. Include a BEFORE sheet and an AFTER sheet in the workbook if needed to show the process you're trying to complete or automate.

Make sure your desired results are shown, mock them up manually if necessary.

Remember to desensitize the (remove confidential) data.

To Attach a File:

1. Scroll down to the window below your post Additional Options
2. In the frame Attach Files you will see the button Manage Attachments
3. Click the button.
4. A new window will open titled Manage Attachments - Excel Forum.
5. Click the Browse... button to locate your file for uploading.
6. This will open a new window File Upload.
7. Once you have located the file to upload click the Open button. This window will close.
8. You are now back in the Manage Attachments - Excel Forum window.
9. Click the Upload button and wait until the file has uploaded.
10. Close the window and then click Submit.