I'm with you so far
One of the things I've been aware of is that the bases need to be placed in locations that are accessible (be it via parkor, or simply walking), which is why I've always envisioned a game where the bases are defined by the players, rather than randomly via the game.
If you did was random bases, I guess the above could be mitigated against by setting up a boundary (the edges of the park) and dead zones (bodies of water, etc..). I think something like this is more of an implementation detail than anything fundamental to the base idea of the game however.
This is an interesting idea, and may solve the issue of 'people looking at their phones' as mentioned by @mavit - you capture a base by performing a physical task. This could be, like you mention, a particular route that you need to take to reach it, or maybe some other type of physical activity, like jumping jacks (which the phone's accelerometer should be able to recognise and calculate)
I love the ideas, keep them coming.
The only thing I'd mention in addition to the above is liability. I don't think it could be promoted specifically as a parkor game, as you just know some idiot is going to go jumping off a building and hurting themselves. I get that people could just as easily hurt themselves running around a field, but this is far less so, and we can potentially mitigate against this with the usual warnings about being safe.