Why is the gravitational constant.. constant?
We don't actually know that it is. Check out the Dirac Large Numbers hypothesis:
"According to Dirac's hypothesis, the apparent equivalence of these ratios might not be a mere coincidence but instead could imply a cosmology with these unusual features: The strength of gravity, as represented by the gravitational constant, is inversely proportional to the age of the universe: $G \propto 1/t$. The mass of the universe is proportional to the square of the universe's age: $M \propto t^2$. Neither of these two features has gained wide acceptance in mainstream physics..."
For myself I can't see how the mass of the universe can be increasing, because conservation of energy is one of the tenets of physics. But I'm willing to entertain a reducing gravitational constant.
Many scientists have now come to the conclusion that a big bang might not explain the 'start' of the universe
That's how it's been for a long time. Big bang cosmology has never allowed us to explain the start of the universe. It just lets us extrapolate backwards by circa 13.8 billion years to a time when the universe was much smaller and denser.
and are coming up with alternatives.
I'm not sure what you're thinking about, but some of the "alternatives" I've heard about don't seem to fit in with general relativity, and don't seem to make any sense.
Could it be that gravity is dependent on the expansion of the universe?
There may be some relation, but IMHO not the way that you're thinking. Einstein described a gravitational field as space that's "neither homogeneous nor isotropic". On the large scale, the universe is homogeneous and isotropic, so there's no overall gravitational field. Hence the universe didn't contract under its own gravity.
Could it not increase as the expansion of the universe continues?
If anything it would decrease, because the gravitational gradient would flatten out. And even it increased, and even if there was some kind of overall gravitational field, gravity alters the motion of light and matter through space. It doesn't suck space in. So gravity is never going to make the universe contract.
This would of course be true in reverse, as the the universe contracts to a near point, gravity becomes repulsive and causes 'a big bang'?
I don't think so. The universe isn't expanding because of gravity now, and I can't see how it ever was.
Is any of this plausible? I've been contemplating the universe for too long. Since our frames do not allow for measuring on a universe scale, perhaps we can't determine differences in the constant?
IMHO a variation in G is plausible, but gravity causing expansion is not.