No, you won't need a public facing collector.
However, if the server which is serving the web application content cannot be instrumented you will need to send the monitoring signal to an alternative web server endpoint which is instrumented and user-facing. Any web server supported by the AppMon agent platform should do (Apache, IIS, Tomcat, Jetty, Nginx, Node.js etc.).
To clarify, I check the box to turn on the CORS option under the UEM application settings and change the monitor request path to point to an instrumented web server ( ie: http://www.google.com/dynatrace/), correct?
This will allow the JS tag of the un-instrumented site to send it's data to the instrumented web server?
Are there unique steps that are needed to make this work?