I think that would actually work very nicely.
A cookie is leakable because the client chooses to send it, so copying it to somebody else is really bad. A server-assigned per-connection ID is not leakable unless you can spoof the IP address of the one you're sending as.
Problem you may have is that some clients are behind proxy farms and can arrive with different source IP addresses within the scope of a single session.
If you do not bind the 'server-assigned-per-connection ID' to an IP address they become just as 'leakable' as a session cookie.
I suspect anyone suggesting that an IP be part of the session security has never actually tried it on a large scale.
An IP address tells people who you are, where you are, and what you do.
A unique identifier ('+18El1iZRFCIiqRpfw4dJR8mXjJn2UxPrjwoRNpjSWg=' for instance) tells what you do without the who and where [1].
It's even possible advertisers could still target users based upon these identifiers, just without the background knowledge that makes these sorts of things privacy issues.
[1] I'll admit, it could be argued that the what can determine the who and where