After quite a bit of experimentation (luckily I had lots of room in to-be-covered areas), Minwax gel stain proved to be the best option. Varathane brand gel stain didn't seem to adhere as well, and Polyshades couldn't achieve a dark enough colour even in several coats. The key was to apply a reasonably healthy coat (or several) ONLY to the peeled areas, carefully wiping any excess off the surrounding areas to avoid darkening the existing stain and leaving a ring. The end result is far from pristine but it does a satisfactory job of camouflaging small imperfections until a full refinish can be done some years down the road.

The entire second floor is the same color as the stairs, so I would like to leave the stairs alone. Eventually, I would like to put a runner on the stairs to minimize the contrast, but need a banister color that would go well with both floor colors.

If I was given the choice I would not carpet the stairs. I would keep the wood as is ( if it is in good shape) or stain it to match your floors. If any money is saved I would replace the metal rails with classic wood spindles.

Looks great too! As a minimum I would paint the spindles (balusters). But this is not a simple project. Very time consuming to do the proper prep, etc. You look like you have less to do than bell-above, but non-the less, a very big undertaking.