I would argue that anything that is going to be 'used' by people, whether they interact with it or consume it has a user experience attached, and therefore there are UX considerations that should be taken into account when the thing is created.
Stairs are no different.
For me there are three considerations that I would take into account when designing a staircase and each is going to have an impact on the user experience:
1. The context
2. The practical
3. the aesthetic
If you look at the two images included in the question, each staircase occurs in a particular context. The primary consideration in designing the fire escape was practical. How do I meet the local building regulations, get people as quickly as possible safely to the bottom of the building using as little space and money as possible. The designer of the stairs in front of the supreme court definitely wanted to impress visitors to the building and so lifted it up on a pedestal. The wide treads of the staircase, the material used and their impressive sweep are all part of increasing the drama of the approach. Each staircase has been designed appropriately for the context within which it is found. Swap them around and they would be ridiculous.
A staircase obviously has to do its job of getting people between two places that are at different heights. It needs do do this well in both directions, up and down (a ladder is much easier to ascend than descend). This consideration has been answered elsewhere, but the idea that a staircase should have a pitch of 30 degrees is not strictly correct. When architects design a staircase they use a concept called 'the going' of the staircase. The steeper a staircase is, the greater the ratio between the risers and the treads should be i.e. the treads should get narrower and the risers higher as the staircase gets steeper. There is a kind of middle ground at about 30 degrees where the 'going' takes the least effort and feels the most comfortable.
Given the context and practical considerations have been taken into account, there are still an infinite number of ways to design a staircase. This is where aesthetic considerations come into account. How should it feel physically as someone walks up and down the staircase? Am I trying to communicate something with the staircase? Am I trying to elicit a particular emotional response? Am I trying to refer to a previous staircase designed by someone else? These questions are endless and are what makes architecture such a great art.