Remove the wall completely, and have the counter extend past the sink at counter height creating a seating are which can double as a buffet for parties. It will make the area more spacious feeling. To get a more comfortable overhang of about 15" you can install a narrow depth bookshelf on the outside end and a cleat of 2x4 bolted to the studs on the wall side. This will take away the need for expensive supports under the countertop material, and will give it a nice built-in look. make sure to put an electrical outlet under the overhang - I find this very handy. To cover the backs of the cabinets, you can order a finishing panel, or, just use drywall painted in the same color as the walls. If there is a concern about flooring material under where the wall used to be, you can just move your peninsula back by 4-6" to cover the spot, giving you a few more inches of kitchen.

Remove it. Once you have the flooring in, if you still want a divider, that can be accomplished with a piece of furniture. You'll have more layout flexibility that way. A sofa table with 2 lamps (or a long, low bookcase) would make a much nicer divider than that wall. Get floor plugs installed before the flooring. I do that with my living room/dining room. During most of the year the DR is rarely used, so the sofa table and couch slide back for a bigger LR (and the DR still looks normal). When it comes time for enlarging the DR table for holidays, we just slide the couch and sofa table forward a couple of feet. Presto!

Yes, you're going to need a new stringer, but more for properly detailing below the treads than supporting what would otherwise cantilever. This is a middling/advanced DIY project. I hope you're up to it, because you're going to get stung if you don't.

We started removing the wall. First half the wall. I was so disappointed of result. Not a lot of changing. I started regret why we even did it. Then we removed whole wall, replaced all wires. Made some arch, big arch. It was absolutely different effect! I love it now. All spare rooms gained space ! More convenient to use!