I don't have the time now to inspect the new circuit, but I guess, I know the result anyway. Removing the DC component with a capacitor is just another word for a high-pass. So, that's one step forward. Without noise added, you should see a full-wave rectified sine without any clipping. I expect that it's O.K. by reducing the level.
With a large noise amount added, you basically see noise at the output. That's normal operation. It seems to me, that you didn't yet think thoroughly about synchronous demodulator operation principle. Viewn simplified, it just shifts the signal around the reference frequency to zero in frequency domain, but it doesn't yet filter any components. So the SNR isn't changed. The purpose of a synchronous demodulator reveals with a low-pass filter at the output. You can e.g. apply a 0.1 Hz low pass filter, so referred to the input, all frequency components except a small band around the reference frequeny are suppressed. The detected signal appears now as a DC value. Because noise power is proportional to bandwidth, you can't remove any noise with the low-pass, but reduce it considerably.
As previously mentioned, also the reference frequency odd harmonics pass the demodulator and detoriate the SNR, so you may want to filter them before entering the demodulator. In principle, a sine reference signal also suppresses them, but for various reasons, the performance is worse anyway.
That's, in a short, the SyncDemod operation principle. Think about it, read some literature and evaluate it in your example circuit.