I agree that, until Clinton, no president was really in the position to run for a third term. However, I think just the possibility of serving a third term would have a big impact VP selection.
After all, the VP slot is often awarded to ambitious politicians to use as a stepping stone for their own bid in eight years, with the tacit support from the president. But if the president isn't necessarily stepping down in eight years, then that changes the dynamic entirely; you wouldn't want to give a platform to a rival to potentially primary you, or who wouldn't go along with additional terms; likewise, a young, eager politician might be less inclined to play second banana if it's for an indeterminate number of years. So for example, Clinton might very well not pick Gore here, and choose a boring elder statesman.
Different VPs would have an impact on successive politics; not in the sense that a different veep would flip an election, but that it would change careers.