> Or is "push-pull" equivalent to "switching to AB"?
No, but in audio it goes-together-well.
You can't pass audio through Class C.
You can't build a single-ended Class B audio amp.
Class C and SE Class B are only good for crude power or with tuned loads.
So single-ended audio amps MUST be Class A.
Push-pull audio amps can also run in Class A. But in push-pull, Class B audio becomes possible, and the advantages of Class B are compelling. Power loss is half of Class A at full power: 3/4 the power iron, 1/2 the heatsink. In speech/music operation, the advantage is greater, since we run below 10% max power 90% of the time. A Class B speech/music amp will run cool 90% of the time, a Class A amp idles at twice the max sine power 90% of the time. And believe it or not, Class AB can give lower THD than Class A (but it is easy to lose this advantage, and THD is a terrible audio metric).
Text-book Class B requires the devices to be cut-off at zero signal. That's unrealistic. In speech/music systems, we never have zero signal, we have a dynamic range. There is always something down at the bottom of the range, and it may be important. Also all practical devices lose gain at low current, so for low-distortion operation we have to bias them up to some minimum current that gives the same/similar gain for small or large signals. In "small" (up to 6L6) tubes, "Class AB" may be more A than B, reaching cut-off only on the largest peaks. In BJT amps, good linearity may be had with idle currents less than 1/10th of peak current. In almost all normal operation, the devices go to cut-off (or a standby bias) pretty much all the time (each half-cycle except the softest passages).
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> I didn't state my design goals ... I'll add them now:
At what gain??
You show unity-gain, but then you mention "2520 footprint". IIRC, the 2520 has a HIGH impedance inverting input, for flexible gain setting and relative indifference to feedback network impedance. This plan seems to have a low impedance inverting input (assuming you break the unity-gain feedback link). It will "work" in voltage-feedback schemes, but performance will vary with feedback network impedance, which is less true for voltage feedback amps like 2520.
The obvious way to meet your goals, stated and implied, is the good old 990 topology, also seen in Self and probably in the 2520. Yes, it isn't push-pull throughout. As you found, setting bias current is a lot easier if one side varies and the other side is a fixed current source, because overall feedback can force the varying side to balance the fixed current. The diff-pair input stage naturally tends to high impedance both sides and to low offset voltage (indeed if it works at all, it probably works with <20mV offset). You can adopt/adapt the 990 values and meet the 40Vpp 600Ω and 250mA specs; you can simplify the 990's complex compensation if you only work at modest gain and/or don't wish to plagiarize too much.
The Diamond Buffer can work outside a feedback loop and can push somewhat Class AB with excellent linearity. The main flaw against your stated goals is that DC offset is sure to be >=+20mV (because PNP stuff don't conduct as well as NPN stuff), and there is no good place to trim/servo that out.
40Vpp 600Ω on one hand, 250mA on the other hand, begs the question: what does it do between 33mA and 250mA? Do you simply want to keep potential destruction down to 1/4 Amp, distortion not specified, or do you want to drive <100Ω loads cleanly? (And if so, at 40Vpp/2 Watts, or will you take less?)