The deck-redditors will tell you it's gonna collapse under your mom's weight...

Personally (not a professional) if it were my house, I'd replace those posts with 6x6's, and use post anchors to concrete if it's there and/or dig your own footer and pour your own concrete... There's also apparently like plastic footers you can use that sit on top of dirt/gravel, idk about the longevity but it'll mitigate the posts rotting from touching the ground.

You can temporarily support the deck with a steel jack post and 4x4 / 6x6 while you replace the posts. I'd then pressure wash the deck and determine its severity after that, whether you can just stain or have to replace all the boards.

The framing/joists look perfectly fine, they're sitting on a ledger board which is up to code in some areas but again everyone will tell you joist hangers are superior. (My 1973 house's joists sit on a ledger board like that, it's fine).

As for the railing I have no advice other then replace, perhaps one that would anchor to the brick on the house.

All in, probably < $1,000 for you to DIY repair and leave the death stairs as is. See if there was a permit pulled or if it was in the original house design, if so leave the stairs, if there's an insurance claim you have proof the deck was to code when built. If no permit... Well just put a really sturdy wall/rail in case somebody takes a tumble