If the codepoints differ from client to client, then using them in chat becomes pointless since the symbol one client types may become another symbol for someone else. You could always use a system like on Discord where emojis are represented by a custom format, but at that point you lose the simplicity that having a dedicated codepoint would bring.
Inline images should undoubtedly be possible, I just don’t see a benefit of being able to register private use codepoints over using something else like rich text for them.