They filter the signal by putting a shield of radio-absorbing material under the antenna itself. Unlike your phone, hand held, or removable car one, the orientation of the receiver is a given and such techniques can be employed.
Non-survey grade GPS positioning is a code-based lock, not carrier, so there is no phase correlation that I am aware of.
Not all differential signals are ground based, the geo-sync WAAS satellites are the most common free source of differential signals (outside mobile phone applications)(at lest in the west, and there is no common terrestrial differential correction elsewhere in the world), while there are also many commercial differential sources, most all are based upon satellite relays.
As to your worry "that people may have not thoroughly considered what may go wrong", I feel those fears are founded (no offense) on ignorance. The system proposed here would replace only the primary layer in a multi-layered system. A replacement which not only closes more risk vectors than it opens, but shifts the risk factors which do exist further from likelihood.