This is a very interesting question that I spent much time studying. Newton never used nor invented a gravity constant. The formula for gravity force he states by words corresponds to $F=Mm/r^2$, so without any gravity constant. It is then of great importance that Newton had a very different view on matter (and mass) than modern physics. He explicitly states that the philosophy of all his work is rooted in that matter consists of indivisible particles with spatial dimensions. Newton and other scientists used Newton's work for hundreds of years to do long series of predictions, such as relative masses between heavenly objects. Newton even calculated very accurately the density of the Earth relative to that of the Sun. What Newton struggled to do was to find the density of the Earth relative to a known uniform substance. The first one to do this accurately was Cavendish in 1798. But also Cavendish did not use nor invented nor measured any gravitational constant. What is true his measures as well as a Cavendish apparatus can be used to find the gravity constant, but that is first after one re-define mass in terms of kilogram.
Newton mentioned mass and weight are proportional to each other as long as the objects of interest are at the same distance from the center of the gravitational object. This is true also.
The gravity constant was first mentioned in 1873 by two French physicists. This I think is not coincident. The gravity constant is needed when one re-formulates mass as kilogram. The kilogram was invented in France just before the beginning of 1800, but was first accepted in scientific circles around Europe around 1870. In 1873 they used notation f for the gravitational constant, some soon used k. People in 1894 used the notation G, but it took many years before G became the standard notation. If one uses f, G, or k as a symbol, it is naturally just cosmetic. What is important that no one used the gravity constant before 1873.
I have tried to summarise some of the history of the gravity constant in a recent article (not in a top journal), but still perhaps of interest as I many references to work there that people themselves can look up and make up their own minds history and use of gravity constant.