270 will work great, goats can take a surprising amount of punishment and run a long way if you don't hit them properly.
17hmr is shit on goats, well with 20 grain hollow points anyway... how do I know? I had to finish off 3 of them that had been wounded with a .270. Head shots were exploding on the skull. Not nice.
I have shot them with everything from .22lr to .338WM. They can be scrawny beasts so the important thing is relatively fast expanding projectiles( on bigger calibers especially) that can open and avoid pencil hole wounds. Hornady 150grain SSTs are now my favorite for .308.
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