To anyone with the same SMS texting problem I had with my new Verizon phone, who finds my post in the Verizon community forum, you’re welcome.
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. [Rather than a Verizon contractor store.] His default Android messaging app does not have the problem with lost messages when the phone is off
Follow-up: The answer is that the LTE (4G) network needs to be enabled at all times. Wifi alone is not enough for all of the apps. Without LTE, some things work and some things don’t with the default Google Android phone and texting apps. In addition to the problem described above, voice message notifications come in, but you can’t access the messages without resorting to the old method of dialing *86.