On a whim I decided to put a LILYGO T-Display S3 into the mix and see if it didn't behave better. The code shouldn't require a change between hardware platforms thanks to the hardware abstraction layer in the Arduino development platform.

I hit upload and POOF, worked first try. There is something amiss in the Pico implementation of WiFiClientSecure or some other related library. I wanted this to be a Pico project, but an ESP32 project is just fine.