Amplifier auto shutdown design
The 741 is powered from 0 volts and +12 volts. This means that the maximum positive output voltage is restricted to about +10 volts (on a good day). This voltage is then fed to an emitter follower (Q2) and the maximum output voltage on the emitter will be about 0.7 volts lower than the base hence 9.3 volts.
This voltage is further reduced by D6 to maybe 8.7 volts and, the final transistor (Q4) emitter output will be lower by another forward diode drop. I'd expect to see about 8 volts feeding the relay on a good day.
If your relay is 12 volt rated on its coil then that is why it likely doesn't operate.

Problem is, the voltage on the relay is very low and doesn't pull.

This doesn't surprise me.

Is my design flawed

Yes it is. Here are other problems: -

The 741 is a really poor choice these days and it has been for a couple of decades.
The input signal you are feeding in is biased around 0 volts and the lowest supply on the 741 is also 0 volts and this is just beyond the capabilities of this dinosaur and many modern op-amps. You just can't expect any op-amp to work like this.