Yes, that's possible. Economic prosperity does reduce birth rates, and that gives me hope. But it's a race between that and the carbon emissions that will happen on our current trajectory towards that prosperity. Right now it looks to me like the carbon is going to win.
The population thing is already on a trajectory toward leveling. The main cause of population growth now isn't birth rate, but rather increased lifespans of those born two or three generations ago. If we convert to a renewable energy base before population levels off, we win.
The thing about politicians that can be bought, is that someone else can buy them, too. If they're not ideologically committed to the cause, if it's just political, then they'll turn on the cause in the name of either ideological goals or simple practicality.
There's a hard limit to how much harm a politician is willing to do to their own country in order to placate special interests with nothing to offer but money.