You do not need to download a new public key in HTTPS frequently (except if you use forward secrecy, in which case a mixed scheme could be used), most people vi

You do not need to download a new public key in HTTPS frequently (except if you use forward secrecy, in which case a mixed scheme could be used), most people visit only just a few specific sites. In addition public keys could be cached. Anyway, a lot of web pages are already much bigger than the McEliece public key.