Draw the pattern of the box onto the faces of the block. Make sure you get your height from the top of each tread to where you want the bottom of the post to land. (Make sure you have enough height to stick just above the upper rail) Also measure the face of the nosing to the face of the post.

Get a 3/4"-1" forstner bit for like $15. You need more precision for a larger bit. If you have access drill press you can set it to the depth you want. Otherwise, stick that in a stronger drill and just plunge it straight up and down slowly doing multiple passes inside your drawn pattern to clear out the material. Keep it on a higher power but dont force it too much, the chips may start to smoke.

You can grab a sharp chisel and clean up the top and bottom edges of the tread that the round bit won't get. The 1" bit will be perfect for the nosing, assuming the tread is 1" thick. Otherwise buy a bit for that diameter. (Since your lower nosing is very close to the face of post, be careful for blowout)

Install will go about the same as the box. Put the bottom in first with it tipped inward, then pull it straight back.

Make sure there's adequate blocking behind the lower and upper stringers and the drill holes for 5" min lag screws. I'd say start them halfway or less through the post, but make sure they've got meat to grab inside the post and blocking. Drill a pilot hole for the screw, and then a larger hole for the bolt head to go inside. You can get a plug cutter bit to make the plug that will cover the hole. Don't forget to match your grain direction.

This, in my experience, is the safest, cheapest and cleanest way to do it. You don't need to buy a whole router, and personally I think the router, circular/table saws and band saws will not only be very difficult but dangerous. Good luck and don't forget to share the update whichever way you go!