Why did the Estates General fail to solve France's problems?
It's hard to say for sure, but a very likely reason the Estates General failed to restore order to France and prevent the Revolution from continuing is that the Estates General was not very representative of popular opinion, being heavily slanted in favor of the interests of the rich.
We often talk about how democracies are slanted toward the rich now, and this is not entirely baseless, given the way campaign contributions, lobbying, etc. work; but at least in France and most other countries today each person's vote actually counts the same regardless of their income. This was not true of the Estates General.
The Estates General was broken into three sections, the First Estate, representing clergy (about 1% of the population), the Second Estate, representing nobility (about 2% of the population), and the Third Estate, representing... everyone else (about 97% of the population). Each Estate got an equal vote, based on the result of its internal votes; so someone from the nobility or clergy respectively had a vote that counted 48 or 97 times as much as someone from the general population. It was common for policies that 95% of French people wanted to be overruled by the 5% who didn't, provided that most of those 5% were nobility and clergy. And all this, of course, only if the monarchy willingly convened the Estates General in the first place, which they almost always chose not to do, only convening it during times of crisis. Before it was convened in 1789, it had not been for over a century.
Even beyond that, the convening of the Estates General in 1789 was intentionally slanted to silence republican voices within the Third Estate, to make it seem as though the unrest was just a handful of radicals rather than a broad swath of public opinion. As a result, the Estates General was widely considered illegitimate and unrepresentative of the French people, and did little to quell the desire for rebellion against the monarchy---and thus, violent revolution occurred soon after.