You'll need to have Visualforce debugging enabled in order to see your QuickAction execute from your Visualforce page since that's where it's called from, not from an Apex Controller. Sending an Email is actually a form of DML. I'd look for DML execution in your debug logs.
If you really want to see the results of your QuickAction, I suggest you look at the methods in the QuickActionResult Class. With it, you'll be able to use methods like
isSuccess() which you'll see in your Visualforce debug logs.
The link you've provided is to the Defaults Handler Interface. All it does is set default values for an instance of a Standard Quick Action that you're essentially "customizing" to suit your specific requirements. You're not really creating a "new class of quick actions". Instead, it's an instance of the
SendAnEmail Action. I'd generally only expect to see where the QuickAction is called on the page and depending on your debug settings, you may not see your class' debug statements appear at all. A unit test is where I'd expect you to be able to easily see your debug statements executed.