The problem you are likely to have is in attaching to something solid. When someone grabs a stair rail it must hold them from falling in any direction. Factoring in the leverage they have from a rail 36 inches above the surface and you have a tremendous lever arm and a lot of force at the attachment point. Wooden posts need to be relatively large (3x3 net) to work as supports for a handrail, and need to go into the ground at least half the height above the ground or be fastened solidly to masonry using expansion anchors. The anchors need to be spread out enough to create a solid attachment. 6-8 inches apart is typical. Other methods include driving 2 inch angle iron into the ground and then bolting the wood to the angle iron (through fasteners, not lag screws). Steel may also be mounted in holes in the masonry and the post bored to match. The metal would be at least ¾ inch square tubing (just like a metal post). I would consider using about 1 inch diameter round steel rod mounted into bored holes in the stone and substrate (you need at least 6-8 inches) and then a mating hole in the bottom of the post for at least 10-12 inches deep. The picture you have does not comply with modern codes since there are too many steps to not have a handrail (though local interpretation and codes may allow this setup for exterior landscaping steps). If it is actually entering a house you will need to comply with the code (no landscaping escape).

One question: does the center spindle (not sure what it's called but the one that's fatter than all the others in the middle of the swirl at the bottom of the steps) - does that get painted too or should it match the railing?

I've gone through this on my old house, and yes it does require a railing by current code. However, in most cities you are not required to bring it up to current code unless you are doing a renovation of that area. When you install the new railing, in most cases the first step is to use a piece of 1"x4" lumber, cut to the right length, and screw it to the studs just below railing height, and parallel to the baseboard. You could also add trim to this piece to make it match a baseboard design. This new board then provides a secure place to bolt any brackets for the new railing, and you don't have to match brackets up with studs. Now find a railing you like, possibly one with lower profile brackets than the standard ones at the home store, and bolt it to the 1"x4". You might also try using something like a pigs ear handrail, and may be able to bolt it on the wall without the 1"x4", which would get it closer to the wall. Here are some examples. https://www.pinterest.com/pin/98234835605074441/?lp=true Bruce

The post I mentioned would not be at the bottom tread so the design of a continuous handrail would have to bypass the post to get to a newel at the bottom tread and there would need to be a guard at the new opening whether it be solid, balusters, etc. At first glance it seems a potentially messy solution. Perhaps the handrail could be at the other side and the guard could terminate at the post. The bottom tread should have had returns at each side. Its looks like a abasement stair in the entryway.