It would be expensive though. This is one reason why I consider 1024-bit end entity certificates much less of a threat than 1024-bit CA roots.
While you could use a faux CA root to sign faux certs for any site you want (ideally ones who are customers of that CA), in practice your use is severely limited. If faux certs are spotted and no one knows where they came from, suspicions are going to be raised. Not only is your faux CA root compromised, but now you may have tipped your hand regarding your capabilities.
To limit that possibility, your attacks would have to be extremely targeted. The more often a fake cert is used and the more people exposed to it, the higher the likelihood that someone will notice what is going on.
It also doesn't help you decrypt the real traffic to the site, or historical traffic, which busting the site's actual SSL key can yield. This presumes that you have a way of intercepting said traffic, but I think it's pretty clear that that is not out of the question (public wifi / ISP cooperation / fiber optic taps / malware). It's more work to bust individual certs, but you're leaving a smaller trail and you aren't sending out examples of your RSA cracking capabilities to your opponents over the public Internet.
Lowering the risk of exposure will let an attacker use the same methods over a much longer period of time, which I think is the goal here.
As to how to combat this: there is a lot of low hanging fruit. Besides the obvious, I would love to see much shorter expiration times for certs become the norm (as in weeks, if not days). For this to realistically happen in a widespread fashion, at minimum CAs need to embrace the concept from a pricing perspective.
With public funding, lots of hardware and expert math/algorithm experts, it's less expensive
That is, even if the generated key-pair is really 1024-bit strong (and doesn't have any biases known by them)