TL;DR Put in the Handrail and Be Happy It Isn't Worse
A handrail is clearly a good idea on a steep set of stairs like that. If it was only 2 or 3 steps, no big deal. If the steps were each much longer (3' or more) then no big deal. But based on the picture, you need a handrail. Not necessarily for code (code mostly deals with inside the building) but for safety.
The first time you try to climb up/down those stairs carrying something heavy in one hand, so you can't steady yourself so easily with the other hand on the bricks, you will really appreciate being able to grab a handrail.
The first time an elderly relative or guest tries to climb up/down those stairs, they (and you) will really appreciate being able to grab a handrail.
If an inspector really wanted to give you trouble, they could also point to:
- Difference in height of the bottom step.
- Possibly (I can't tell from the picture) height of each step and/or ratio of length of step (run) to height (rise). I wouldn't be surprised if one or more of these dimensions is out of the usual specifications for residential stairs.
If the inspector really pushed, they could cause a lot more problems than just "add a railing". Are they right? Maybe, in which case there is the question of why the problem was not spotted earlier in the process. Are they wrong? Maybe, in which case it is a power play (if I don't find at least 3 things wrong at each place I go to, I'm not doing my job well).
But in the end, it doesn't matter. This is not a battle worth fighting, because a railing here is actually a good idea.
(My electrician has told me of stories where he fought the inspector and was able to cite code and win. On the other hand, he came prepared to add an extra ground rod that he knew my jurisdiction didn't require, in case the inspector demanded it when inspecting my heavy-up - he knew that the cost and time of installing an extra ground rod (which he didn't up needing - he'll use it on the next job) would be less than what it would take to fight with the inspector. You pick your battles based on what is worth fighting.)