On a true 60 amp 240 volt circuit will each side of the breaker carry more than 30 amps if required?
No, that's not at all how it works.

This feeder will provide 60A @ 240V. That is 60A on each leg of the panel, NOT 30A on each.

The 60A is available, you use what you need up to roughly 60A, again, on each leg.

A residential panel will be hard pressed to be fully balanced, or greatly imbalanced, due to the transient nature of usage.