My grandfather’s house had a staircase just like that, although the steps were solid oak (and much thicker than the ones on the post), and the secret for its stability is that the steps are affixed to a brick wall and have a complex set of metal anchors inside the wall, it’s not as scary as it sounds, you just have to avoid climbing it while drunk.
The staircase i’m describing is also quite resistant, as we moved my grandfather’s turn-of-the-century, solid wood desk and the steps did not buckle or crack, even after 50 years of being there.