In chicago, at Illinois Institute of Technology, they had a park in-between the residencies and classrooms; when they decided to build a main center (McCormick Tribune Campus Center) at the park, they (Rem Koolhaus) tracked the pathing that students were naturally taking to cut through the park, with the assumption that these were naturally optimized routes to important destinations. The building was then constructed such that those pathways persisted. This worked out quite well, in usage. (except some of the administration doesn't actually grasp this concept, and arbitrary block off routes... most often for their security circus)
>whoever comes first and is part of the majority who wants to travel from A to B through the park without casually strolling will define the way
I'm not sure about whoever comes first; people don't normally follow paths based on lightly treaded footprints in the grass. Hell, you watch people cutting across the grass and you're not going to see them just line up.. They walk to their destination, and enough people to do so over the same path (presumably, because its optimal, or close to it), and the grass dies, and a pathway naturally emerges, and then you get your lines. It wouldn't be about which person arrives first.. it would be about which destination appeared first. The pathway would be naturally created for it, and then if a different, but nearby destination appeared, they'd probably re-use the pathway rather than to create a new one. A sufficiently distant new destination would likely create more pathways though.
>What if the park is supposed to have a few loops and not-directly-diagonal paths to encourage lingering around?
There's nothing stopping that; just don't interfere with the optimal paths. There's no reason you can't have more paths (it's not like you're putting fences so they can't intersect..), but certainly its unreasonably to have fewer paths than what people will naturally use.
Also, what is this supposed to business? It's a design/aesthetic choice, and if its unreasonable to have given the conditions of the environment, it would be utterly moronic to put it in anyways. If the designer doesn't have the authority to not be subservient to whatever is supposed to be, then its likely the stupidity of the bureaucracy. This then isn't question about what is good, its about what the authority thinks is good, and by how far its willing to force it onto the population.