Being able to crack present-day communications in the future is still a concern.

Hence https://en.wikipedia.org/wiki/Forward_secrecy to make things more difficult

Compromising the main keys isn't enough, need to compromise each session key as well in turn, a massive increase.

Forward secrecy does not provide any value against cryptography compromise. Quite the opposite as it depends on the security of the cryptography over the long term to insure old messages stay inaccessible.

The session keys are exchanged by the asymmetrical systems that the imagined quantum computer would be able to break. So the attacker gets the session keys directly.

I mean, it depends a little bit on what your threat model is. If it takes a week to break a key, and you have hundreds of thousands of tls sessions without knowing which is the relavent one, it is definitely something. But yeah it seems like it would quickly become a minor hurdle once real quantum computers become a thing and presumably have their own moore's law.

I'd imagine that's exactly why the NSA hoards the (rumored) absurd amount of data in the Utah data center