In my new house these two rails:

are wobbly and feel unsafe. It seems to me that whatever I do to the existing pieces, they're always going to be rather unsafe—they're low (less than 3 feet high) and will always have a tendency to get wobbly rapidly, due to the one-ended way they're attached and the small area that's contacting the walls. .

I'd like to replace them with something sturdier—ideally something to which I'll then be able to attach a top-of-stairs baby gate.

My first problem is that I've been unable to even search for replacement items, because I'm at a loss for how to describe these top-of-stairs-sawn-off-stairwell-side-rail-ettes. Can anyone tell me what they're called, point me in the right direction to shop for replacements, and/or supply any further advice on how to construct something better in the limited space available?