By design Tableau Desktop does not color dimension values because dimensions values are considered headers and not actual data. The goal is to get the end user to focus on the data so they can more quickly see trends and outliers.
I am unaware of any additional workarounds besides using the container fields that I already see in your screenshot.
The Tableau website has an Ideas section where customers can add, comment, and vote on features they would like to see in the product. Votes are used to help prioritize new features. Our Development team also loves to see everyone's use cases in the comments, which helps them better design features that will meet the most possible needs.
I found the following Ideas you may be interested in:
Hope this helps
Thankyou for your response, saved a lot of my time. I will surely notify this to add new feature in the product.