Look, they don't actually need the public onside, do they? The public have no power in the sense you mean it - as some nebulous democratic force - to prevent climate change. Even if millions of people cared deeply about climate change, we would be stuck with the relative apathy of the two main parties. The parliamentary system just isn't set up to give us a Green majority in Parliament in the 2022 election. It took 30 years and a World War for Labour to go from a good idea embodied in a social movement to having their first Prime Minister in a minority government, and it took 30 more years and another World War for them to have their first majority. We can't afford to have 30-60 years of upheaval before we do something on climate change. That's the whole point of the protest.
Extinction Rebellion's demands are all aimed at the government directly. I'm not saying it is likely that the government will fold after a few days of - let's be honest - pretty tame disruption. But I suspect plenty of people in government would have at least visited their website now. Maybe if they keep doing these protests, month after month, it'll get more difficult to ignore.
And why should green energy only be implemented when it is so-more-profitable than fossil fuels? Come on man, the planet is more important than some faceless megacorp's profits. The government can force them to be green -- but only if we force the government to do it.