Rob, You got some solid advice here and hopefully you found a solution that suits you. I agree with DA: My take on it is that the crack is going to increase (or even crack somewhere else) even if you glue it or use a plywood patch on the back, glued and screwed. And then is the problem of how it will look like. Considering that your stairs are new, you wouldn't want it to have a crack already. What I would suggest is using a string cladding product to hide your stringer: