You are absolutely correct, the time measured by such a clock is going to be dependent on general relativistic effects, most prominently by distance from the mean geoid. However, I fail to understand how you jump from that to concluding that it's useless. For example, you could use such a clock to make precision measurements of general relativity and test possible extensions. Moreover, a clock that sensitive should be able to "feel" changes in gravity caused by density fluctuations in the Earth. This could help find oil deposits, for example. The summary says as much.
Generally speaking, you NEVER lose by increased precision. It is true that if your specific application is limited by low precision in some other component, you won't gain by increasing precision somewhere else. However, that's not the case here. I'll admit that I don't know enough about GPS and satellites to answer your specific question, but my impression is that they currently ARE limited by time standards.