Is it possible not to have $_GET values in views' exposed filters? I can't seem to find a setting to do this nor am I familiar with any hook to implement this. Is it even possible?

I don't want to use ajax though.
You can enable Ajax and extra stuff will not be added to the URL. 
If the problem is too long URLs, do a google search and see how short the URL is. However you can clean up the URL (that means you will use $_GET values) by setting a shorter URL parameter identifier. 

Click on the exposed filter and expand the exposed filter settings. You will see a text field to enter the parameter string for this filter. Give a meaningful but tiny string. 

For example, default URL would be  like:
path/to/view?field_my_weird_field=drupal

You can change it to:
path/to/view?text=drupal