Very odd craftsmanship, indeed! Depending on your budget, you could perhaps open up the entire stair case to make the open-concept design option would work. I can't see the entire space but from what I can see, it would be nice to remove those walls and have some open stairs with railing that match all the way to the floor and re-tile the entire entry so you don't have to hunt down the exact tile. Like this:

first off, hire a tile setter who knows how to tile.

Depending on the budget, there are a few options. Find a reputable tile person.

tile should look similar to the way it's laid here. chair rail piece edging does not belong here!

retile using a curved nosing.

rip it out, replace w/same wood from stairs

redo stairs and entire downstairs w/new tile. this is slate, outdoors, but indoor could be done in similar fashion. much neater.

if you like the stained concrete look, rip up all the tile and have a cement overlay done. stain it and have the bare concrete.

BTW, measure those stair treads and risers. they don't look up to code. If you do buy this house, and those stairs are not up to code, that is something you can negotiate.

This is just weird and not to code. For good reason. This is a trip hazard. Too small to be a landing to big to be a tread. So, don't just redo it in wood. Re-build the bottom two steps to match the rest of the stairs. If there is room you can do a radius on one or both ends.

I believe that this may have been inspired by the classic interiors of the movie 'The Holiday'... However in this application it just does not work

http://1cqgxm3l59yi2wwbnn3qy35h.