"HTTPS Everywhere!" means "Heartbleed-like bugs everywhere".
There are several readily available TLS stacks for embedded systems (CyaSSL, PolarSSL, etc.) and plenty for other platforms (Open/Libre/BoringSSL, NSS, etc.), so 'people without a working TLS stack' is not a real-world use case you need to take into account.
Requiring TLS forces people to include TLS stack, which enlarges trusted computing base a lot. Security depends on many things, but the size of trusted computing base is an important factor.
That's a silly argument. Here is how I will rephrase it: don't use computer because computer software always has bugs. Millions of servers run on all variants of Linux distro and every month we find a dozen of security vulnerabilities. Even standards have bugs and unresolved items.
Most of which only apply if you have local untrusted users or are running particular software.
You can give up if you like, but I still believe it's possible to set up secure systems.
Try writing one of those from scratch and auditing it. (No, seriously, actually do try, break out Rust, do some proofs; more secure code would be awesome.)
EDIT: If you exclude sctp, the remaining IP/UDP/TCP/ICMP parts are 1/8th of the size of OpenSSL.
What about the HTML 5 parsing, the CSS parsing and layout, and the JavaScript virtual machine? I doubt you could write and carefully audit all that in a weekend.
Most sites are already inaccessible to people who don't have modern web browsers, which all include a TLS stack.
But specifically about your point, when one audits an HTTP/1.0 client, should one also audit the TCP/IP stack in the kernel? I don't think most researchers/engineers would, and would (for sake of practicality) instead trust the underlying systems. Eventually, TLS will be thrown into that "underlying systems" bin. That time is not likely at hand; there are still too many shortcomings of TLS and its surrounding infrastructure. As evidenced at least by the OP. But when the time does come, writing an HTTPS client in a weekend will be just as practical, since you'll trust the software libraries/kernel to handle TLS. Just as you trust your libraries/kernel to handle TCP/IP today.