It could, with IndexedDB or similar, but then you have a massive conversation history being stored on user devices. Space is a precious commodity on phones.

For text? Nah. Most apps automatically store photos on your device, and that's fine. Mobile phones have several GB for storage. That is enough for hundred thousands of messages, before even noticeable. Please reconsider your E2E policy and don't make the world a more insecure place.