Whether in Europe or US power is the same in that they all use redundant ground. The first ground is through the neutral wire that completes the circuit. The redundant ground is earth ground. Millions of homes here in the US are wired with the older single neutral ground. Not a problem.
RCA cannot work without ground. The outside is ground, the inside hot, break either one and no circuit no sound. You did the right thing. There's more expensive solutions, highly recommended too by all the guys around here with money to burn. If that's you, go for it. Otherwise, relax, pat yourself on the back for finding a creative solution that cost almost nothing.