Since the 1970s, TAI construction has had to compensate for the differences between the physical locations of the atomic clocks in laboratories around the world and an ideal surface of equal gravitational potential around the world.
Nowadays, BIPM and other laboratories routinely talk about general relativistic corrections across the width of the measuring devices. To quote Appendix 2 of the SI Brochure:
> In 2013, the best of these primary standards produces the SI second with a relative standard uncertainty of some parts in 10^16. Note that at such a level of accuracy the effect of the non-uniformity of the gravitational field over the size of the device cannot be ignored. The standard should then be considered in the framework of general relativity in order to provide the proper time at a specified point, for instance a connector.
Being able to measure the drift at a specific point doesn't mean it's relevant to computers timestamping their calculations. If a computer is ten nanoseconds off, that's basically the same as it being one rack to the left, or having some slack in the cable. There's no real effect.
Just syncing your clock once a day is enough to let you completely ignore the effects of relativity.
Even between stars the difference isn't that big, but offsets don't matter at that scale anyway. When it takes a decade to send an email from one system to another, it doesn't matter if their timelines are offset by a week.