Only large merchants are able to maintain such an infrastructure; (even
Coinbase recently failed at it, they forgot to update their
certificate). For end users that is completely unpractical.
The same benefit can be achieved without the complexity of BIP70, by
extending the Bitcoin URI scheme. The requestor is authenticated using
DNSSEC, and the payment request is signed using an EC private key. A
domain name and an EC signature are short enough to fit in a Bitcoin URI
and to be shared by QR code or SMS text.
bitcoin:address?amount=xx&message=yyy&name=john.example.com&sig=zzz
That extension is sufficient to provide authenticated requests, without
requiring a https server. The signed data can be serialized from the
URI, and DNSSEC verification succeeds without requesting extra data from
the requestor. The only assumption is that the verifier is able to make
DNS requests.