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Even if we build a spaceship that can accelerate up to the speed of light and send it to the nearest star, we won't even know what happened to it in our lifetime. Doesn't seem like something the world could agree to donate all its money for.

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People talk about how long space travel takes, I feel like people forget that at 1g of acceleration the entire galaxy is within 22 years of travel.

For an observer in the ship, obviously. Here, outside, it will be more like 200 thousand years, and constant 1g acceleration is impossible.

You are also forgetting how hard it to maintain that 1g of acceleration for a long period.

Just 1000 years for a stationary observer vs 22 years for a astronaut under 1g constant acceleration. It's not that radical.

The diameter of our galaxy is about 100,000 light years. Unless you've got some way to go faster than light then you're 2 orders of magnitude out on your estimate.

That's ignoring relativity! For the astronaut, travelling at near C, dilation will occur and it will seem to be less time.

Aside from many other technical challenges (including how to shield your super-fast ship from microasteroids so it doesn't get holes punched in it at 0.5c), the main problem is: where do you get the energy needed to maintain constant 1g acceleration?

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