I think lighting should be purposeful and I don't see a reason to illuminate those areas. A light at the bottom of the staircase would illuminate a few steps and the surrounding floor. Why?

For the staircase, I'd consider putting a ceiling light above the staircase or a wall-mounted, downward-focused light on the wall above the staircase that faces the staircase door.

-- amateur

Lights inserted into that 5" wide "soffit" would serve no purpose other than lighting a strip where the wall once was. It would certainly highlight the flooring patch job yet to occur.