> what value does paying with your phone add over an Oyster

Phones and contactless payments actually do the fare-capping calculations slightly differently - if you do a "commute into London, then around London a bunch, then back" travel pattern, it can purchase a virtual central-London travelcard in the middle of the day, and adjust the commute-in payment retroactively. If you want that fare treatment with Oyster then you need to get the travelcard option loaded on it in advance.

> or contactless

modestly better security