Hi there,
Yes i see your design which looks like a roll your own design wont work
without a phase shifter, you are very right about that, but i dont see
what is so lame about using an op amp or two? There is a little more
that goes into a high quality amplifier like this than meets the eye, and
some of that has to do with the amount of distortion on the output.
Using op amps, you can control the transistors pretty well.
If you want to stick with transistors then perhaps you can try just
building an inverter (that's 180 degree phase shift) with a gain of 1
and use that to drive the lower section. The trick is to get the gain
to be 1, and have it stay that way even if some change in ambient
temperature. The top gets driven the way it is i guess, and the
bottom gets driven with the inverter.
I cant say that this thing is going to sound that good though, but at
least you will be on your way to a better amplifier.
You also have to do a quick check on quiescent bias current to make
sure it is not enormous, which a simulator wont alert you to and
in real life would draw wayyyy too much current to be feasible.
The way to beat crossover distortion is to keep the 'off' transistor
actually turned on a little, even when it is supposed to be off.
The trick here is to make sure the off current is high enough to
keep the transistor in the linear region but not so high that it eats
up power and heats up the transistors.
It's understandable that the circuit doesnt work as is, because the
transformer is not being driven with the correct phases. Once this
is corrected you might have a workable design in progress.