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.