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But we didn't send the probes from the Sun.

Plus I don't think it got its velocity from Earth, but by using gravity assists to steal momentum from other planets (which, if you don't know how that works, do yourself a favor and read the wikipedia page, it's pretty great).

Remember that we’re already orbiting the sun though. A lot of the hard work is already done.

The premise is about current chemical propellants and structural components. If you build a rocket out of nanotubes with nuclear propulsion you could go wherever you want from anywhere else. Except black holes.

You have to have high thrust to get out of the dragging atmosphere.

Sure, we could orbit Jupiter and get back out. But if we ever descended to the "surface" we couldn't climb back out.

Voyager 1/2 and New Horizons are at escape velocity and leaving the solar system.