Were that true, the theory would never have survived general relativity. It isn't true, and the theory does. The gravitational component of the energy is not enough to prevent the expansion, that's what general relativity tells you. What you are close to, however, is a second issue that is indeed very puzzling-- why did the energy partition itself between expansion and rest energy (using the terms loosely, GR doesn't use energy like this) in such a way that the expansion neither went "thud", nor occurred too fast for anything interesting to happen in the short times allowed? Both of those possibilities seem vastly more likely in a "blind" universe, so ours seems to be a very carefully selected universe. It is puzzling to find a scientific explanation, though nonscientific explanations abound (the anthropic principle, religion, etc.). This isn't a problem with the theory, it's a mystery of the universe. This is simply the usual misconception that the BBT is an explosion. It isn't, and your picture would fail on many levels, the most notably being that it would give rise to spatial gradients (where did the "collision" first occur?) that are not observed. So the problem has nothing to do with gravity (your picture would certainly overcome any gravitational binding issues, but as I said, they were never an issue anyway), but with fitting observations.

I'll let you be the judge.