finishing hardwood stair treads at top step

[Tigercubrider]

I am looking to change my stairs carpet to aftermarket add on hardwood cladding.

There is parquet downstairs, carpet on the landing, the landing carpet comes down the stairs.

I would fit the treads myself. There's a half round step at the bottom and I would use white risers either by painting the original riser or covering with thin sheet material.

The question is that the landing carpet currently overhangs the top step, and comes down the first riser, where the "actual" stair carpet starts. There is a piece of timber at the top of the staircase that is obviously part of the stairs, rather than the landing floor. The strip has a half round edge and matches the width of the landing newel base.

I presume this is replaced with the treads? and the landing carpet would stop there?
(rather than overhang and look weird from the hall?)
I mean that I would take a tread and cut it to the depth of the original piece?

thanks