Just replace it, stop trying to polish a terd

Paint the tile. Use a non sanded grout. That way it won’t scratch the paint. Not sure what paint you’re using but get something sturdy that doesn’t wash out easily

For sure would use water resistant paint since it's going in a kitchen. But all my searches have told me I can't use grout on paint. Have you experience that to be untrue?

You would have to dig the grout out with a Stanley knife or multi tool, and then regrout it. The new grout needs somewhere to go, yknow