Using the default WooCommerce shipping option needs to be configured this way.
In this example, l will be using our WooCommerce USPS shipping plugin and the Flat Rate option in WooCommerce.
The same rules can be used for all our shipping plugins.
The priority option for the USPS is 1
The priority option for the Flat rate is 2
Here is the default option admin area
Next, you will need to go the USPS admin area and select this option
Select shipping rate to display: Cheapest
You need to select the option Cheapest rate if you would like USPS to display as the default option on the checkout page.
If more than one shipping service is enabled for use, and you do not select the option cheapest rate, USPS will never display as the default option.
About the WooCommerce Local Pickup option.
*Please note that if using the WooCommerce Local Pickup option, these rules do not apply, the Local Pickup option is free and this will always be offered as the default regardless of any priority you have set up.
Unfortunately, this is a WooCommerce function and this is not negotiable.