Can the Hegel H300 outperform the Ayre AX-5?

I'm guessing it would be a matter of personal preference, and not that one is better than the other.

The Hegel is very good but not any more of a giant killer than a number of other really good integrated

Personally I would take a Devialet 120 to either of these. The Devialet is way more advanced in terms of technology and sounds amazing.

Or if I needed a lot of power to drive demanding speakers then a little known American Integrated from CODA the CS iB which is a 400 watt into 8 ohms, 800 watts into 4 ohms, and the first 5 watts is Class A. Rock solid build quality from the original Threshold engineers.

I've heard both and it's without a doubt personal preference in your environment. I actually ended up with a Hegel H200 over the Ax-7 and Ax-5. I heard both the 7 and 5 sound better than my Hegel, but not at my house with my other components. The Hegel worked better for me. I needed the extra power. Both amazing companies that produce giant killers IMHO.