Can you display data dynamically from a textfield in the URL and Name of Views?
I think you can still use taxonomy for this. The taxonomy can be open ended not need to make it closed to the terms you define. 

When you add a taxonomy reference field choose the widget as autocomplete this will enable the user to add their own taxonomy term which may not be there in the list already. 

If you are very particular about using the field then you can use that a contextual filter in a view so that it would appear in the url as mentioned.