Hi, Mel.
Thanks for your detailed question.
The “next” link only gets hidden if there aren’t any future events, since we wouldn’t want a visitor to click infinitely to the next month to never find any events.
I see your site has targeted .tribe-events-nav-previous, .tribe-events-nav-next to add a colored border. I think you should target .tribe-events-nav-previous a, .tribe-events-nav-next a instead.
I think this would be more along the lines of what you’re wanting to have appear.
Please let me know how this goes for you.