Lawrence,
The symptoms we are having do not seem to match your explanation.
All messages/notifications that are sent do make it to my iPhone and display on the lock screen, regardless of how much time has passed with the phone being off. My iPhone shows me the number that sent me the message and the text/content of the message.
BUT, when I unlock my iPhone, some of those messages (mainly the ones from Android users) never show up in the Messages app (and also disappear from any notification history).
If I check online from the Verizon side, it says that the text messages were RECEIVED. It doesn't say anything about not delivered.
The other day, a friend (Android) had sent me 4 text messages when my phone was off. Once turned on, I saw all four on the lock screen, but when I opened the Messages app, only 3 showed up. The missing one was the 4th message he sent, the earlier 3 were there. This does not match with your explanation.
Also, I don't think this problem has ever happened to me with an iMessage from an iPhone user, so what you say about not mattering if it is iPhone does not appear to be correct.
Never have any of my friends ever told me that they received a 'not delivered' error when sending me a text.
Saying to never turn off your phone for fear of missing text messages is a very poor solution.