I've never thought about a Primer style of time-travel for a game. Bill and Ted just always seemed like a more grokkable way to do the idea. But it sounds fascinating.
I wanted to respond to this with my thought process. When I first looked at the question, I tried to think of what my touchstone works were for Time Travel, and I came up with four pieces of media that really stood out:
Back to the Future,
Doctor Who,
Steins;Gate, and
Puella Magi Madoka Magica.
As a warning for anyone who wants to watch any of those, ESPECIALLY the latter two, this discussion is going to get into spoilers.Two of those options featured bodily time travel where you don't really cross your own time stream. Heck, in
Back to the Future the struggle is to AVOID changing time, rather than attempting to change it. The Doctor, for his part, is usually an observer, at least when it comes to established historical events. By contrast,
Steins;Gate and
Madoka both feature protagonists that are making an active attempt to change something, usually in a fairly immediate sense and about their own experiences.
Now how would any of that time-travel relate to gameplay?
In
Back to the Future, the time-travel is a singular event. Well, a two-of event if you count the fact that there's a return trip to worry about. Similarly, while
Doctor Who may explore all of time and space (Especially alien worlds and distant futures), each 'story' largely stays contained in one setting. The TARDIS arrives, the meat of stuff happens, and then the TARDIS leaves. There are exceptions, but that's the majority pattern. In a game sense, you'd probably be playing out a particular story, or the 1955 core of
Back to the Future. The players might be fish out of temporal water, but the time-traveling is done when the game is set up and packed back into the box. Or, to be charitable, it would happen at very specific points where the game changes 'phases', allowing you to take some stuff from one time period into another. Essentially, Time Travel would be the frame of the game, not the meat. In a way, you could say that
Seven Wonders does that, since each round clearly represents a different period in the development of your ancient culture, except for the fact that the "time travel" is strictly forward in nature. And there's nothing wrong with that, I think you could, if you wanted, have a good time travel game built like that. But you'd have to work on it and ask yourself what makes it different from a game that was just set in the "meat" time period with no travel element?
But then look at
Steins;Gate: this is a story that features three distinct kinds of time travel and practically countless time-travel events.
Madoka is more contained, the bulk of the story taking place over a single timeline, but the time-travel element introduces itself with a similar multiplicity. It's actually very similar to the second act of
Steins;Gate, except the character who is tied in to the time travel isn't the main character we're following. Translate either of those to a gaming experience, and time travel has to be a mechanic. I'd probably focus on the Time Loop/Time Leap side, since human players are temporal beings and will be experiencing the game without interruption by their future selves (As the looper/leaper has a consistent memory), at which point the key elements are repetition and iteration. Board Games are already pretty good at repeating and iterating behavior, since recurrent structures give them some consistent identity, you'd just need to define what elements reset, and what changes remain in place to let the players progress towards their 'endgame'. In that way, the Time Travel would be fairly integrated into the gameplay experience. The key notion would be learning and prediction, skills that, again, board games are already good at rewarding.