I tested it with your demo account and have only one comment:
When I select a different carrier than the default one I am taken right away to the payment selector. That's really nice.
However, if I keep the default one I have to click "Select payment method" to be taken to the next stage. It's the same for the payment selector. Would it be possible to not have anything preselected so that the customer must make a selection and is then taken automatically to the next stage of the checkout? Otherwise the customer has to click eg "Select payment method" to be taken to the next stage. I'm not sure that's so intuitive, but I'd like to hear from others here in the forum what they think.
Apart from that, it's super nice!