Just like any empirical constant (like 'little' g), G is the attempt of humans to fit a value to a quantity that has no real significance. G is just the empirical constant Newton needed to describe the effects of gravity. On a macro scale, knowing a 'constant' to within 1% is pretty damn good. Since Newton's equation for gravity is a macro scale equation, that's fine for me.

For example, when calculating the gravitational attraction between Sol and Earth, the masses of the bodies is less well known than 1%, so the error added in by G is marginal at best.

Most of the constants of nature are known to much better precision than G, it has been notoriously difficult to measure accurately. It's not really a scandal, just a well-known difficulty. It doesn't affect the truth or falsity of any theories.

Wouldn't it effect many of our accepted astronomical measurements though? I realize the formulas remain unchanged but the output seems like it might be significantly different.

Cosmological constraints on variation of G over time are much tighter than those established based on solar system experiments. Here is an example based on WMAP data:
"WMAP constraints on scalar-tensor cosmology and the variation of the gravitational constant" http://arxiv.org/abs/astro-ph/0311274