Can you display data dynamically from a textfield in the URL and Name of Views? I have a content type of Product which includes a text field for the user to enter a Brand. This text field is open to the users to enter correctly, as there are hundreds of brands and I don't want them to have to learn how to enter taxonomy terms.

I am wondering if there is a way to dynamically create a View that will also dynamically generate the page name and page URL depending on the brand, and then display a table view of all of the Products with that Brand.

So for example if I have /category/brand/product-name as the parameters for the URL, for a specific product it would be /tshirt/busted/open-road, and with Busted being the Brand entered by they user, the view would automatically create the /busted url, and populate it only with Busted brand products in a table view.

Is this possible? I was thinking perhaps I needed to use tokens to grab the [node:field_brand], but I am unsure exactly how I would do this, or if it was even possible. Also, if there is a better way of handling this data instead of a textfield, I would like to know!
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.