[Square] The payment form is not apperaring until selecting a shipping method

Yes, I have noticed this issue too, but the problem is that I do not know how to make this issue steady reproducible: it is completely random for me.
I suspect it could be an issue in the Amasty One Step Checkout extension.
If you know how to make this issue steady reproducible (on every page load), then I will be able to debug it.

As a workaround, you can set the Square payment method as the default payment method in the Amasty One Step Checkout settings: