I'm not sure what you mean by "emojis come from the rendering application". Some applications (particularly web applications) forcibly replace emoji characters

I'm not sure what you mean by "emojis come from the rendering application". Some applications (particularly web applications) forcibly replace emoji characters with inline images because they don't trust every user-agent to have a sufficiently-complete emoji font installed, or because they want to use their own custom emoji designs rather than the platform-native ones. That's not a fault of emojis or the font setup, though - that's just paranoid developers. Most UI toolkits will render emoji characters according to whatever emoji font you have installed, just like they'll render Cyrillic characters according to whatever Cyrillic font you have.