You could train the system prompt into the model. This could be as simple as running the model with the system prompt, then training on those outputs until it had internalized the instructions. The downside is that it will become slightly less powerful, it is expensive, and if you want to change something you have to do it all over again.
This is a little more confusing with Anthropic's naming scheme, so I'm going to describe OpenAI instead. There is GPT-whatever the models, and then there is ChatGPT the user facing product. They want ChatGPT to use the same models as are available via API, but they don't want the API to have all the behavior of ChatGPT. Hence, a system prompt.
If you do use the API you will notice that there is a lot of behavior that is in fact trained in. The propensity to use em dashes, respond in Markdown, give helpful responses, etc.