I don't think you've been promoted beyond your level of competence, I think you enjoy coding more than managerial troubleshooting. There are lots of articles on engineers who prefer engineering more than managing.

Not what you're asking for, but you have a pretty strong hand to play right now. If you went to the CEO and listed the issues that you're dealing with that you would expect the CTO to deal with.

Then say, "These are basically executive-level functions. I'm happy to handle them, but if you aren't planning on hiring a CTO, I'd like the title of VP Engineering and another 20% raise."