You're right that SNI is part of TLS rather than HTTP. It's the same trouble though. Most applications don't support TLS+SNI.
The way Tor does it is to use a SOCKS proxy. Then you don't need an IP address and it's protocol-agnostic but the client application has to support SOCKS.
I'm not sure address space limitations are even a major problem. Address assignments are local, not global, and there are millions of RFC1918 addresses.
For HTTP you have to solve it from the other side anyway. An HTTP server would be more likely to run out of addresses than a client would. But running out of IPv4 addresses is what IPv6 is for. We could even return both IPv4 and IPv6 addresses until the IPv4 addresses run out. Then if you want to burn through millions of peers you just have to support IPv6.
The way Tor does it is to use a SOCKS proxy. Then you don't need an IP address and it's protocol-agnostic but the client application has to support SOCKS.
I'm not sure address space limitations are even a major problem. Address assignments are local, not global, and there are millions of RFC1918 addresses.
For HTTP you have to solve it from the other side anyway. An HTTP server would be more likely to run out of addresses than a client would. But running out of IPv4 addresses is what IPv6 is for. We could even return both IPv4 and IPv6 addresses until the IPv4 addresses run out. Then if you want to burn through millions of peers you just have to support IPv6.