Do you have customers from EU? If yes, I do not recommend to change payment button. In embedded flow it would be misleading and breaking regulations.
What do you mean by synchronization label? I will take it with devs.
Thare are payment methods that rely on custom buttons PayPal, Apple/Google Pay so IMHO there is no generic solution that all payment methods could have same “Place Order” label. Even our payment button is not default Woo order button it is our custom button and we replace default button because we had many incompatibility issues with 3rd party plugins. In the future our payment button will be very likely bellow card from (different place from default order button).
I will investigate our options hooks/API that could have long term support but at the moment we have no API that we maintain.
I will keep you updated