To rewire or not to rewire
The old wiring in your house appears to be old-style AC (BX) with cloth insulated wires in a paper overall wrap under the spiral armor (no bonding tape).  Its fault current capability in the event of a ground fault is limited -- the armor in this early BX, while grounded, is not up to current standards for an EGC (equipment grounding conductor).  I would check the newer wiring to make sure that the EGCs are continuous to the main panel, and fix all the miswired outlets for that matter (i.e. reversed wiring on outlets).

You could replace the ungrounded outlets on the BX circuits with what are known as self-grounding outlets if you wanted some sort of ground connection on them instead of none whatsoever, but I'm not sure that's essential with modern levels of GFCI and AFCI protection.