I took my circuit to a simulator called TINA-TI and now I understand. The mid-supply biasing doesn't work, as current flows out of the opamp and into the potential divider. This destroys the middle bias and sends the output to the upper supply rail. If the input signal is large enough, it dips into the middle and then the output works as expected with a huge amount of gain.
I put Audioguru's circuit in and it seems pretty solid except two things.
I did an AC analysis and the gain over frequency is pretty poor. The datasheet says that the TDA2030A has an open-loop gain of 80dB at 1kHz. The data entry in TINA doesn't mention a frequency so I entered 10k. Would that be right? As far as I can tell, this low open loop gain is what's holding back my high frequency response.
I made a few tweaks and managed to shift it over a bit:
However I'd like to know if this is either a mistake or something that can be improved.
Secondly, this arrangement will still have one of the flaws that my original design did: lots of DC output when initially turned on. As the input cap charges (and the others, but they seem to have less effect), the output goes to 10 volts for a small amount of time which may damage any lesser wattage speaker that I connect.
Decreasing the value of R6 (Audioguru's diagram) lessens this spike but also negatively affects the bandwidth and gain. Same goes with the resistor closest to the input.
Time delay relay on the output maybe?