Space

Why would it cost more energy to accelerate the same load on a linear path than on a circular path?

I didn't say that. I said that you would need to input a lot of energy in much less time.

Now, how would you do that in a linear acceleration system? Well, you would need an unimaginably large capacitor bank if you're going the electrical route. Or, you could use chemical fuels, but then stuff would get way more complicated as you would have to deal with plumbing (if you're using fluids), reliable detonation, etc.

In this case, you're just storing all that energy in the carbon fiber hand, which basically is a flywheel.

And why do you assume the time has to be minimal? You can make the rail quite long, kilometers long in fact.

All acceleration that a payload would receive is when it's in the gun. So let's say, the first 100m give it a 1000g acceleration. But, when it enters the next 200m, it's already going very fast. Therefore, it would spend very less time in the next 100m section. Thus, the payload would get WAAAAAY less acceleration.

This means, that you get diminishing returns in terms of initial velocity as your gun increases in length.

Another issue is that you would be able to launch stuff in one direction only. What if I want to launch in polar orbit? I would need to build another km long gun in that direction. Compare this to spin launch, where you could quite easily point the launch apparatus in whatever direction you want to launch in.