Ted, there are ways to do what you want which would "work" as Billy Bob and Bubba will tell you, but to do it safest and especially IF the NEC is of any concern or adopted or required in your Jurisdiction THATS ANOTHER STORY.
To begin with, while perhaps Romex or non metallic sheathed cable may be proper for residential home and stud wall wiring methods, a single insulated non sheathed wire may or more likely MAY NOT be permitted in your jurisdiction. Normally, (how I was taught and in every jurisdiction I practiced) the Ungrounded Conductor and the Neutral Grounded Conductor and the Equipment GroundING Conductor ARE ALL CONTAINED IN THE SAME RACEWAY OR JACKET OR ENCLOSURE.
THAT ALL BEING SAID (cant help myself) to your more specific question. YES splices should be in an enclosed readily accessible junction box (cant have them alone in the wall or in a hidden box DUH) and if the box is metal, it also needs bonded to the Equipment GroundING Conductor.
TRUE the Equipment GroundING Conductor is usually NOT a current carrying conductor as its intended to ONLY carry Fault Current. HOWEVER that's when its the most important and valuable and may be required to conduct a total short circuit fault and carry high current and develop heat even though only for a short time until the breaker trips IF ALL IS WELL AND IT WORKS AS DESIGNED. Often theres only smaller currents in the normal wiring but Fault current is often HIGH so you want such a conductor in a protected proper safe environment. UNDERSTAND the wiring may be rated for only 15 or 20 amps BUT A DEAD SHORT CIRCUIT AT 120 VOLTS WILL (until breaker trips) CARRY MUCH HIGHER CURRENT AND CAN DEVELOP HEAT AND/OR MELT THE INSULATION. True its the same size of wire as the others, Im just saying the enclosure or raceway or wiring method or cable type is very important even if carrying fault current only.
So for Billy Bob and Bubba and before anyone has a calf, YES it will work,,,,,,,,,, NO it may well NOT be the safest and NEC (if adopted or required) approved,,,,,,,,,,,,FOR SURE make any splices in an accessible Junction Box IM SURE YOU ALREADY KNOW THAT
NOTE there are a lot of current practicing trained professional electricians and engineers here (plus plenty of lay Billy Bobs with opinions and methods that will "work") so see what they have to say as I'm pretty rusty on the latest codes and practices.
NOTE they make plastic flexible conduit into which individual wires can be used, that may be one wiring method to consider. The thing I can’t say is what’s required where you live or what wiring methods are permissible IM ONLY SAYING RUNNING INDIVIDUAL WIRES NOT KNOWING THE WIRE OR CABLE OR ENCLOSURES OR JACKETING MAKES ME NERVOUS as I was taught and practiced having all wires in the same enclosure or jacket etc.
Your intention is good as 3 wire receptacles provide more safety, just be sure to do it safe and proper, but stringing individual wires in an existing home that aren't in an approved raceway or enclosure or part of a multi wire assembly would raise red flags in my rusty but well intentioned opinion.
Do as you like, not my call, but please keep safe
PS Good Sparky man drsporster raised a good question about using GFCI’s.
John T