Two testing cases: (server and installation are the same)
ee site create mysite1.com --wpsubdir
Manually installed w3tc plugin through Add New Plugin option in the WP dashboard and - what a suprise - it function totally normal, based on the test you suggested with the timestamp.
It really surprised me because w3tc rewrite rules are in the nginx.conf at htdocs folder and it should not work with the EE configuration.
ee site create mysite2.com --wpsubdir --w3tc
Everything OK, but I understand that page cache works with the pre loaded EE configuration (wich is perfect and nice but not for all cases) and whatever I change in W3 page cache option not gonna have effect.
ee site create mysite3.com --wpsubdir --wpfc
Everything is perfect, but I’m sure you have a bug in your Nginx-Helper plugin. It doesn’t purge (automatically) all the post and pages when you create a new post when the plugin is set to do this. You should do it manually with the big button.