I confirm this seems like a bug and I am able to reproduce it.
To be more clear about the problem, if event calendar pro is on the home page, the menu links for the views are not shown if the url is like
if the url is
/home it works as it should.
I will investigate this and fix it for the next update to Event Calendar Pro.
One separate advice:
What I do usually instead of putting Event Calendar Pro directly on the home page create a separate page "Events" to be the main page for ECP.
Then on the home page add an instance of Event Calendar Pro - List View named Upcoming Events, point this feature settings to the Event Calendar Pro that is already on the events page so it will show data from that one.
However list view does not show the links to the other views so it does not solve the problem you are asking about. But I find I usually don't like to clutter the home page too much so its best to have the links only show once they get to the events page, at this point the user has shown he is interested in events.
Choosing what page to put Event Calendar Pro on is important because if you move it to another page later after it has events it will break the friendly urls for events which have the old page id.
But that is just the way I recommend, you may really want to put it directly on the home page, its up to you.
I will fix this bug for the next update and I will post on this thread when that update is ready.