When the null Unicode character appears in chat, the game output shows the error
<log4j:Event logger="fin" timestamp="1723129374266" level="INFO" thread="Render thread">
<log4j:Message><![CDATA[[System] [CHAT] ]]></log4j:Message>
</log4j:Event>
XML_ERROR_PARSING_CDATA
An example of how you can reproduce this:
Run command
/tellraw @s "\u0000"
The game output now shows the error shown above.
Confirmed in 23w46a.