I did not check the numbers since I’m not that familiar with US units and load combinations, but it does not look like it should be that unstable.
I have two things to say:
1) I would not generally consider the help from a sidewalk unless I really need it, I’m certain that it will stay there and not be removed and that it can transfer that force to some other element.
2) could you add a tie on the right side and connect it to an existing structure (it could even be a concrete slab - I have not done this, but I feel that it should work)
I imagine you are using Rankine earth pressure to check the stability of this wall. Rankine substantially underestimates passive pressure. Instead, try using log-spiral theory for the passive pressure at the embedded portion of the wall. This may not solve your problem entirely but you'll gain a lot of resistance to overturning.
That soil weight, is that something that can be "repaired" during the excavation? 120 psf is relatively dense, if sand at 70 or 80 pcf would help? It seems like this has to be all excavated out anyway, then you're hoping they recompact it so it doesn't cause the adjacent (upper) structure foundation to shift?
I'm not convinced this is going to math out, regardless of whichever soil pressure theory you drag into the daylight from 1920. That upper structure is also probably producing some overburden pressure on the proposed wall.
I don't see how a 4" sidewalk is going to offer enough sliding resistance to act as a pin, there. All you have is weight and friction, unless the sidewalk slams into a basement wall on the left side out of the picture. OSHA, last I checked, wants a 1/2 slope unless they do something fancy, so both sides need "something" during excavation and construction. Not that OSHA is your thing, strictly, but it has implications for the design.
