Can one use the Apache Traffic Server and not have to use Apache Web Server?
To start with, the "Apache" name is there to show that it's under Apache Foundation control, as it's stated on the web site: "...Formerly a commercial product, Yahoo! donated it to the Apache Foundation, and it is now an Apache TLP."
Also, from there - this is not a web server, but a proxy server.
So, to answer your question - to serve web pages you need Apache Web Server.
If you need proxy, you can use either Traffic server, or another proxy server like Squid.