Hard-core is decidedly iffy. I think you need to have something at ground level to anchor the steps - anchoring into the wall as you have proposed is a weak way to fix the item due to the potential for fastener pull out
I think you need to have it set up so that the top "step" is more like a landing, is deep enough (at least 600mm, preferably 900mm) and us at the exact same height as the threshold. This is so you step up without any risk of a trip and have a "rest point" before descending to garden level
There is a drop to the right side of the current steps so TWO hand raild are an absolute must. Part of the reason you need two handrails is because the customer is elderly and likely as not is mobility impaired, or will become so over time especially if they have arthritis
Have you calculated the going on the stairs to ensure that you meet the minimum legal requirements (arguably these are notifiable works, not sure) and furthermore to ensure that your client can actually manage the steps? (The elderly often need easier going)
Have you considered durabilitu and maintenance? Steps can become very slippery when in an exposed position. You may need to consider nonslip coatings for the treads
Do you have both PL and PI insurance to cover you in the event that the client has an accident? (Essential)