Yes it does. Why won't it? The expanding universe solution was mathematically derived from Einstein's gravitational equations just a few years after Einstein published it. So yes, obviously Big Bang expansion is undoubtedly allowed by gravity. The rigorous derivation is shown below.Scientific conspiracy? Or you could call it misuse of the public trust.
I noticed you did not address the points, how does gravity allow a big bang? Oh, it doesn't, right. Now that's a problem, isn't it?
Solving the Friedman Equation
Cosmological Physics
Thus gravity is perfectly consistent with Big Bang expansion of the universe as the math formally proves.
Also the detection of the long predicted gravity waves for which this year's Nobel Prize was awarded shows beyond doubt that Einstein's GR is the correct theory of gravity for the post Big Bang conditions of the universe.