Continuity doesn't tell you anything about solid connections or current-carrying capability. It tells you that SOME (unknown amount) of current is able to flow through. It doesn't tell you how much. For example, you can test the continuity of a piece of 14 gauge wire. Then test the continuity of a 4 gauge wire. You'll get the exact same results. Does that mean a 14 gauge wire can carry just as much fault current as a 4 gauge wire? Nope.
To test wire and connections, you need to actually put the wire under full load, or close to it. If a wire is supposed to be able to handle 30 amps and you want to test it, you need to actually put 30 amps through it. It's just like plumbing: If a pipe is supposed to be able to handle 60 psi of water, and you want to check it for leaks, you pressurize it to 60 psi and then check for leaks. You don't put only 5 psi in it and assume it's good because it's not leaking at 5 psi.