Sure you can and I agree the railing should be on top of the stairs, It would just take some time to do it. You have to move the 4x4s on the left railing which extends the railing on the deck which means you have to replace a few deck boards. Also I don’t know how the railing 4x4s are attached to the stringers in the deck, might be an issue there, might not. If he moves the 4x4s make sure they are still attached with a threw bolt and not just screwed into a 2x?
F82D89D3-C0D2-4672-BBC3-A19401F51C2B (resized).jpeg
Also you will have to figure out how you want to terminate your skirting around your stairs as the bottom 2x4 will be higher and on the inside of the skirting. I would probably cut the skirting off at the stair and run the tread of the stair over the skirting.