THE PROBLEM: Verizon delivered a Pixel 4a. When the phone is powered off, the default Android messaging app (with blue icon) fails to retrieve stored SMS messages. When the phone is powered on, back-and-forth texting works fine. SMS store-and-forward is broken only when the phone is off.
THE FIX: The Verizon Message+ application (with red icon) does not have the problem. Stored messages are retrieved when the phone is powered up and the app is opened.
RECOMMENDATION: Verizon should make its own MSG+ app the default for SMS texting.
NOTE: My son's Moto G Stylus was purchased unlocked and configured with a SIM at an authorized Verizon store. His default Android messaging app does not have the problem with lost messages when the phone is off.