I'm providing an update for anyone that runs across this issue in the future.
The issue ended up being due to trying to copy over messages with protected strings (info towards bottom of this page):
https://wowpedia.fandom.com/wiki/UI_escape_sequences
An example of a protected string is that message in chat that occurs when a Battle.net friend logs on or off (if you have that option enabled). It was causing all of my copy frame messages to disappear because I am using a single string with line breaks.
Solution:
I couldn't find anybody that was able to figure out what the issue was so I started looking through GitHub repos to see if others were doing something special when copying over chat messages to a frame. Credit to
https://github.com/tukui-org/ElvUI as that was the repo I referenced, but there are plenty of other repos with similar functions.