If you want to restrict yourself to the earth's crust, that'd still be ~5,000,000,000,000 kilos of iron you could use per star, or 1,500,000 Saturn V's. And that's just using one planet.
Sure, there are rarer metals. But I'm not seeing evidence that the solar system wouldn't have enough of them, and mass mining and manufacturing seems like an far easier way to handle them than individual probes (the probe would have to land on the various planets, manufacture refueling centers to refuel itself to travel around the planet doing geological survey and take off again, build mining centers when it finds the deposits, etc.).
The fact that you're talking about Saturn Vs and landing on planets and taking off again shows that you're considering the wrong paradigm entirely. If you're landing on planets you're already doing it wrong. Atmosphere and gravity wells are the enemy. Remember, even the non-self-replicating probe still needs to be self-repairing, since in your scenario it has to last for probably a million years' worth of radiation and micrometeorite damage. A million years' worth of spare parts is probably a much bigger investment than some automated manufacturing.
I brought up Saturn V's to give people a sense of the mass we're talking about, not because I thought the probes would be actual Saturn V's (and I don't think we'll be sending 1.5 million of them to a single star).
As I said, the reason I talked about landing on planets is because you mentioned rare metals, and I don't know how _reliably_ they could be found if you're only looking at asteroids. But maybe that's not an issue.
The problem wasn't that you thought they would be actual Saturn Vs, the problem is that you think the mass of a Saturn V is remotely enough to send a probe to the other side of the galaxy. Designs for intra-galactic probes typically include meters of shielding because of how nasty hitting even a grain of sand is at interstellar velocities, and in your case this is even worse, since said shielding doesn't have to stand up to a century of ablation, it has to hold up for a million years. You're exceeding the mass of a Saturn V on shielding alone, and the tyranny of the rocket equation still applies. Building bigger means building everything else bigger.
Sure, there are rarer metals. But I'm not seeing evidence that the solar system wouldn't have enough of them, and mass mining and manufacturing seems like an far easier way to handle them than individual probes (the probe would have to land on the various planets, manufacture refueling centers to refuel itself to travel around the planet doing geological survey and take off again, build mining centers when it finds the deposits, etc.).