I see a few problems with this approach:
1)Not all content is provided over both HTTP and HTTPS. For multiple reasons, one being performance. Which leads us to the second problem...
True, which is why HTTPS Everywhere only enables HTTPS on sites that support it (they are specifically whitelisted by the extension devs).
2)A HTTPS session incurs a significant overhead for encryption. Which may be no problem for someone like Google. But for someone hosting his/her own (moderately successful) website on a small server, it might just overload said server.
While HTTPS does incur some overhead, it's surprisingly small for modern servers. Google, for example, was able to add SSL/TLS to all Gmail connections with no new hardware, no additional servers, and SSL/TLS accounts for only about 1% of their CPU time (see here for details).
Pretty much any server will reach other bottlenecks before the slight overhead of SSL/TLS becomes an issue. Using Perfect Forward Secrecy is important for security and using DHE-based ciphers do incur a moderate overhead compared to non-DHE ciphers (a factor of about 3). Using ECDHE instead makes the increase in overhead only about 15% rather than 300%. See here for details.
3)Quite possibly the biggest problem with HTTPS is the fact that users have been trained over many years to just click "accept/install certificate" on self-signed certs. Not knowing that if you do this you are no longer secure.
And the more we keep forcing HTTPS, the more webmasters will use self-signed certs. Not many people want to go through the hassle of obtaining (and maintaining!) a valid SSL certificate for every single website they run, even if that cert is free. Which will only exacerbate the problem...
 Getting a domain-validated SSL cert from publicly-available CAs is the work of a few minutes and, as you point out, often available for free or very low cost. Many hosts will automate the generation of a private key and CSR, making the process one of copy-paste for the customer. Other hosts handle the entire process of generating a private key, getting it signed by a CA, and configuring things correctly.
Sure, some sites use self-signed certs, but these are usually for personal or internal corporate purposes and not for the general public. The scary warnings in browsers aren't likely to go away anytime soon, so I doubt that any webmaster of a website meant for public use is going to be using self-signed certs (other than those catering to specific, tech-savvy audiences).