Given that one can get a CA-signed certificate for free, and that in any case the cost of a cert is dwarfed by the cost of running a nontrivial website, I think you overestimate the number of sites which would be "secured" in this manner and the gains to be had.
But the important point is what we'd lose: If a UA accepts a SSC for an arbitrary website, then the NSA can actively MITM a website that uses a CA cert -- the browser will never see that CA cert, so it doesn't know better.
The only way around this would be to accept SSCs but treat them as no different from plain HTTP in the UI. But now websites have no incentive to use the certs in the first place, further limiting the benefits to be had.
Really, just buy a certificate. It's not that hard.
> That would be a major and totally free improvement in privacy for everybody.
Tangentially, I feel like you're trying to have it both ways in arguing for more encryption on the web and also against mandatory encryption on the grounds of energy efficiency. SSCs wouldn't be "totally free" under your model -- we'd be spending carbon on it. I might argue that if one is going to pollute to encrypt their data, it would be unethically wasteful to use a SSC, which comes at exactly the same environmental cost as a CA cert but offers much weaker guarantees.