To do it with a pair of clocks, there's a long way to go, as they measure differences in depth in a gravitational potential. If you change your distance from Earth's center by a meter, your gravitational potential will change by 9.8 m^2/s^2. If somebody heavy (100 kg) and spherical (I'm a physicist) stands a meter away from you, your potential will change by 7 x 10^-9 m^2/s^2. In short, the clocks need to get ten million (they can see a 1 cm height change) times better to sense a nearby person. Clever trickery with an ensemble of clocks will make that easier, but not by 10^7.
Not impossible, just hard. The fact that the time/frequency teams have encountered gravity, in particular through the gravitational potential, has just made their lives quite a bit harder. They now need to know a lot about the relative locations of other clocks and the mass distribution within the earth to make substantial strides forward. It's an incredible feat to have gotten to where they are, and a major challenge for the future.
I wonder, however, if the problems you are describing are things that would work themselves out over time as the equipment improves? The precision of these clocks sound like they're many orders of magnitude better than the old gravity sensors. I think. Perhaps all we'er waiting on is some kind of crazy technological magic over the next 50 years that would involve miniaturization, improved accuracy, and an array of a million or so. (All of which I just made up)