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On your edited data:

Wow, yes, that is pretty incomprehensible. But keep in mind, in the 60s and 70s, those electricians felt toward grounding pretty much the same as electricians today think of torque screwdriver requirements: they think of it as know-nothing lawyers fooling around with stuff that's worked for 50 years. (which is wrong at every level, but that's what they think). Or it could be a DIYer who didn't have Internet back then.

Your plans for working around the broken ground wire seem solid. I've done the "box near the panel" myself on a rewire of a factory, where someone cut all the wire ends at the edge of the panel to steal the breakers. See if you can loosen a clamp and pull a little more wire in from the wall.

By the way, the "box near the panel" is a great trick when you need AFCI on a panel, but the breakers are obsolescent or you just want to pay $20 for an AFCI receptacle instead of $40 for a breaker.

Edit: That panel

It's obvious the dryer guy did not install this panel. It appears to be fed by a 4-wire feed - note all grounds and neutrals are carefully segregated. So this must be a post-2008-style subpanel fed from either another panel, or a main disconnect/breaker elsewhere. (in NEC 2020 style; some cities have required this for 50 years, probably the same reason for the 4-wire feed). So look for a main breaker or main panel elsewhere.

Despite the age, it's not a split-bus panel because it's apparent the panel is all original, and all the 240V breakers have been put down the right side (a best practice that's impossible on split-bus). So, main breaker elsewhere.

If you can't find a main disconnect, all these breakers "rock out" - push further outward on the handle and the whole breaker will unclip from the bus (which is in the middle). Then the heel comes off its little hook and you have a breaker in your hand with wires attached to it. (which is fine; it became de-energized when it unclipped from the bus). This, by the way, is why backfeed breakers must be bolted down. So you can pop breakers off a live panel, change their wires with the breaker in your hand, and snap them back in. Have the breaker be off so it doesn't arc when you clip it on.

Note that you need a torque wrench or driver to set the screw torque correctly. (NEC 110.14). That's serious business that has been traced to many wire burn-up's.

Unfortunately, you'll be doing this a lot. Challenger panels are fine, but Challenger breakers had a VW-like scandal where it turns out they had rigged their UL testing. So they don't trip reliably and must be replaced. The good news is that BRyant inherited the bus design and kept it for their own BR breakers, and their line is made by Eaton. Eaton BR is the most popular breaker line in America, and if you look on the breaker it says "Type BR / Type C" (you need Type C).

So all the breakers need to be replaced, but it's still under $100 for all of them. That's about what you were going to spend on the 2-pole GFCI!

On retrofitting a too-small ground:

This is a universal problem: grounding the 30A dryer. With a universal answer: "NOPE". The wire is too small.

Code is clear in 250.122 that a 30A circuit needs a 10 AWG ground. 15A circuits are wired typically with 14 AWG wire.

14 AWG is smaller than 10 AWG, so it is not a suitable ground wire. In the case you link, that person's #16 ground wire was legal when it was installed, and is now grandfathered because the owner can point to the date on the building permit. Your de novo use of #14 on a 30A circuit is not grandfathered, and has no date because it has no permit.

AFCI is Arc Fault Circuit Interruptor. It uses a digital signal processor to "listen" for arcing on the wires. What you just described is a GFPE (weak GFCI that operates at 30A instead of 5mA threshold). AFCI and GFPE are not the same thing. However, some implementations of AFCI do include a GFPE as a "cheap" way to detect H-G or N-G arc faults (since they are also ground faults). Those would work; however 9 years later when it starts nuisance tripping and you replace it with a year-2031 AFCI, that year-2031 AFCI may not have any GFPE at all. Indeed, GFPE-absent AFCIs are already on the market.

As such, if you want GFPE, use a GFPE or GFCI, not an AFCI.

This is a universal problem: grounding the 30A dryer. With a universal answer: "NOPE". The wire is too small.

Code is clear in 250.122 that a 30A circuit needs a 10 AWG ground. 15A circuits are wired typically with 14 AWG wire.

14 AWG is smaller than 10 AWG, so it is not a suitable ground wire. In the case you link, that person's #16 ground wire was legal when it was installed, and is now grandfathered because the owner can point to the date on the building permit. Your de novo use of #14 on a 30A circuit is not grandfathered, and has no date because it has no permit. (it may not need one).

AFCI is Arc Fault Circuit Interruptor. It uses a digital signal processor to "listen" for arcing on the wires. What you just described is a GFPE (weak GFCI that operates at 30A instead of 5mA threshold). AFCI and GFPE are not the same thing. However, some implementations of AFCI do include a GFPE as a "cheap" way to detect H-G or N-G arc faults (since they are also ground faults). Those would work; however 9 years later when it starts nuisance tripping and you replace it with a year-2031 AFCI, that year-2031 AFCI may not have any GFPE at all. Indeed, GFPE-absent AFCIs are already on the market.

As such, if you want GFPE, use a GFPE or GFCI, not an AFCI.

This is a universal problem: grounding the 30A dryer. With a universal answer: "NOPE". The wire is too small.

Code is clear that a 30A circuit needs a 10 AWG ground. 15A circuits are wired typically with 14 AWG wire.

14 AWG is smaller than 10 AWG, so it is not a suitable ground wire.

AFCI is Arc Fault Circuit Interruptor. It uses a digital signal processor to "listen" for arcing on the wires. What you just described is a GFPE (weak GFCI that operates at 30A instead of 5mA threshold). AFCI and GFPE are not the same thing. However, some implementations of AFCI do include a GFPE as a "cheap" way to detect H-G or N-G arc faults (since they are also ground faults). Those would work; however 9 years later when it starts nuisance tripping and you replace it with a year-2031 AFCI, that year-2031 AFCI may not have any GFPE at all. Indeed, GFPE-absent AFCIs are already on the market.

As such, if you want GFPE, use a GFPE or GFCI, not an AFCI.

This is a universal problem: grounding the 30A dryer. With a universal answer: "NOPE". The wire is too small.

Code is clear in 250.122 that a 30A circuit needs a 10 AWG ground. 15A circuits are wired typically with 14 AWG wire.

14 AWG is smaller than 10 AWG, so it is not a suitable ground wire. In the case you link, that person's #16 ground wire was legal when it was installed, and is now grandfathered because the owner can point to the date on the building permit. Your de novo use of #14 on a 30A circuit is not grandfathered, and has no date because it has no permit.

AFCI is Arc Fault Circuit Interruptor. It uses a digital signal processor to "listen" for arcing on the wires. What you just described is a GFPE (weak GFCI that operates at 30A instead of 5mA threshold). AFCI and GFPE are not the same thing. However, some implementations of AFCI do include a GFPE as a "cheap" way to detect H-G or N-G arc faults (since they are also ground faults). Those would work; however 9 years later when it starts nuisance tripping and you replace it with a year-2031 AFCI, that year-2031 AFCI may not have any GFPE at all. Indeed, GFPE-absent AFCIs are already on the market.

As such, if you want GFPE, use a GFPE or GFCI, not an AFCI.