Sorry, I'm a bit confused about where your line of reasoning is going.
First, it's important to point out that the blanket term of "the left" typically encompasses "liberals" in a neoliberal, corporate Democrat sense, who are not "progressives" by any means.
You blame the "left" for everything such as supporting cap-and-trade and other policies that only benefit the rich and powerful - however, many progressives are already trying to limit the power of the rich elites. See campaign finance reform for examples - they are trying to limit the ability of money to buy influence and power in politics. Additionally, many support emissions trading schemes because it's "better than nothing", which is the alternative at the moment.
It also seems to me that you're attacking the "left" because neoliberalism has consolidated power in the hands of the extremely rich, and therefore any climate action as advocated by the "left" will benefit the very rich. However from a leftist perspective, it's imperative to achieve both goals of a) effective climate action and b) returning government to working for the people, not the elites.
See in 2009 when the Australian Greens (a progressive, environmentalist party) voted to block a carbon pricing scheme because it was not effective enough.
Many progressives actually agree with you that any effective climate action will only benefit those with the money and power to write the rules - and fossil fuel dependence is bad, oil cartels are bad, global warming is bad, billionares and oiligarchs making fortunes from government decisions that benefit themselves under the guise of progressive policy is also bad.
The middle class is slowly being eroded away through wage slavery and declining social mobility, so progressives also want better social welfare programs for the lower class, and more stringent taxation of the ultra-wealthy class so it is *not* the middle class that shoulders the burden of the effort that needs to be done