They are indeed buttons and they should be to help people who are still learning the system but their primary purpose is act as indicators that there's a page that can be found by swiping. Using them as buttons should be discouraged in the current navigation system.
My point is that it should be separate from the dot navigation system. It should not be similar to the old navigation scheme. The tabs would of course function as buttons but the use of gestures would be encouraged over the buttons when convenient.
Perhaps I would understand your idea better if you drew it like I've done. Try this handy tool I used to draw these mockups. You can download it from [this site](https://balsamiq.com/).
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