What application do you have where huge public keys are no problem, but you fear imminent (at least within a lifetime where you can't upgrade) availability of large quantum computers capable of Shor's algorithm?
The arrival of quantum computers doesn't have to be imminent for it to cause harm. The Utah datacenter (and likely others) store giant amounts of encrypted content, waiting for the key exchange to become decryptable. Currently almost all communication we consider to be encrypted is not safe from quantum computers.
How great would it be in 30 years if you had great evidence that people who are in the prime of their careers have done embarrassing things in their youth. You could extort them or use it to get them out of the way if they turn out to be a problem. The value of such information in fact increases over time as those people rise to more important positions of society.
So ideally, we'd switch to quantum confidential key exchange as soon as possible so that such an attack can't be done.
Yes, Facebook does exist. I've written about this before. People who don't have a past with vaguely embarrassing things they did and goofy photos and the wrong friends and so on are not future politicians, they are ghosts and their future is as foreign intelligence agents. Do you know what sank Prime Minister "Call me Dave" Cameron in the United Kingdom? Was it the unverified and likely fictitious stories about a pig - similar to events from a popular a TV show? The family tax scandal? Something he did at school or as a young man? Photographs of him dressed like someone from a different century? Nope.
What sank Dave was assuming that voters wouldn't destroy their own country to spite people they don't like (including him). A depressingly ordinary political tale, that could come from any era in history.
This project that you linked seems quite nice. I especically like its Code of Conduct.
The Nytimes.com front page is 4.66 MB. A Quantum McEliece key is 8373911 bits (per wikipedia) or almost exactly 1MB. Therefore if my connection to Nytimes.com was over McEliece based TLS -- it would be a 21% bandwidth increase. I doubt I would notice.
Even more so if the webpage is loading stuff from many different domains