[m_mueller]

I think you're conflating a few things here:

1.) A probe and a human spaceship are vastly different problems. Since all a probe really needs is electricity to sustain itself, you could get away with a tiny payload and some long lasting radioactive energy source, light sails or even sending the energy from earth's orbit. Such a thing would be either slow and cheap or fast (a few percent of light speed) and expensive, but not both at the same time and I doubt it would be in the trillion dollar range whatever you do.

2.) I completely agree that sending humans would currently not be feasible within a single nation's budget and the technology for that is still at the very least decades out (cryogenics, EM shielding, better propulsion systems, using mass from cheaper solar system bodies than earth etc.).

3.) 1000 years is what we'd need with conventional current technology. The theoretical limit for a nuclear impulse propulsion drive is 20% of light speed if you want to break or 40% for a fly-by.