Electrical – Should I replace the neutral wire that I accidentally cut in a Romex cable when installing a new outlet

Before going through all that I'd check:

1. If the existing metal boxes are already grounded (See How should I connect to a short ground wire in a metal box? and maybe Should old knob and tube wiring be replaced? )
2. If the old cables will come out cleanly from below. Fish a string as you yank out the old cable.

Note:

1. The NEC's unsupported rule does not apply inside walls. You get a free pass on that one.
2. You can put an AFCI breaker on the two wire circuits, no problem.
3. You can put a three prong GFCI on the two wire circuits, as long as they are labeled as such.

That description is OK, but you really only need to run 2-wire with ground romex up to the switch, not 3-wire. Split the hot side of the outlet as described. Use a wire nut to pigtail a short piece of wire from your source hot to the terminal of the receptacle that you want "always on" and also connect to the new black wire from the romex running to your switch. Connect the other end of your new black wire to one terminal of your new switch.

Remember, the switch just "makes" or "breaks" the hot.

Connect the new white wire from the romex to the other side of the switch. Permanently mark the white wire at the switch box with red or black permanent marker to identify it as a "switched hot". Connect the other end of the white wire from the romex to the hot terminal of the receptacle that you want switched (for the disposer). Permanently mark the white wire at the receptacle with red or black permanent marker to identify it as a "switched hot". Connect the source neutral to the receptacle and ground everything.