No, you shouldn’t. Because clearly it’s gotten under your skin, which is a sign there’s some trouble here in your relationship with your co-founder.
But — let’s step back.
You shouldn’t be doing a start-up for the glory of yourself. If for no other reason than that’s not going to work. You should do a start-up for (x) the glory of building something epic, and (y) if you do that — you probably will make a ton of money.
Because of (x) and (y), I’ve found than almost no great founders care at all about public credit per se. Because all they care about is building something great and huge. They don’t really care how it gets done, per se.
But someone has to be the public face of the company, and in the early days, it can really only be one core person. That usually needs to be the founder that can best promote the company and get PR, and, usually raise capital and close the initial customers (usually, one founder does all 3 of these — it’s a related skillset).
Usually one of you, even if you both are terrible at it, is far better than the other.
Yes, it’s a team effort.
But the one that is better at this should be the public face.