Motivated by the benefits of improving our knowledge of Newton's constant G, Feldman et al have recently proposed a new measurement involving a gravitational clock launched into deep space. The clock's mechanism is supposed to be the linear oscillation of a test mass falling back and forth along the length of a hole through the center of a spherical source mass. Similar devices — ones that would have remained in orbit around Earth — were proposed about 50 years ago for the same purpose. None of these proposals were ever carried out. Further back, in 1632 Galileo proposed the thought experiment of a cannonball falling into a hole through the center of Earth. Curiously, no one has yet observed the gravity-induced radial motion of a test object through the center of a massive body. Also known as a gravity-train, not a one has yet reached its antipodal destination. From this kind of gravitational clock, humans have not yet recorded a single tick. The well known reliability of Newton's and Einstein's theories of gravity may give confidence that the device will work as planned. Nevertheless, it is argued here that a less expensive apparatus — Small Low-Energy Non-Collider — ought to be built first, simply to prove that the operating principle is sound. Certain peculiar facts about Schwarzschild's interior solution are discussed here; and a novel way of interpreting gravitational effects will be presented in Part II, together adding support for the cautious advice to more thoroughly look before we leap to the outskirts of the Solar System.