There are several factors that influence the thermal stability, or sensitivity to thermal run-away in that output stage bias scheme.
The easiest to tweak is the emitter "degeneration" resistors. These 5.6 or 6.8 Ohm resistors are supposed to look large compared to the dynamic resistance of the Vbe junction as it changes with temperature and reduce the current increase with rising temperature. It's been years since I looked at the actual equations for this, and it is hard to apply completely because you need to know the actual thermal resistance from junction to air, for the output devices, but suffice it to say making the emitter degeneration resistors larger (say 10 Ohms) will reduce the likelihood of thermal run away. Obviously, perfect thermal tracking between the transistors and bias diodes will prevent the class A bias changing from initial conditions as they all heat up the same.
The trade off in making the degeneration resistors larger is lower class A current for a given DC operating point, and slightly higher losses when driving low impedance loads. Note: things like using different transistor packages that impact the thermal resistance to air and can impact thermal stability and different diodes/devices will have slightly different nominal Vbe/forward voltage.
In power amp output stages, that deal in much smaller value degeneration resistors (.1-.33 Ohm), the matching of the diode drops is more critical. At Peavey we had a dual diode in the system that was specified to meet a voltage @ current spec, and power devices were similarly graded for Vbe @ current.. A lot of work, but worth it to avoid factory trims that humans had to adjust correctly.
If you continue to see thermal problems I would look at increasing the value of the emitter degeneration resistors. If this starves the output stage for class A current you can tweak this some by increasing the current through the D3 diodes. Note: the fact that this is called D3 instead of Dx and Dy suggests to me that some original design used a proper dual diode. Note: Dedicated dual diodes are not always made by stacking two simple diodes. While this doesn't matter a lot, it looks like the design was dialed in for a dual. If the output stage isn't clean with larger degeneration resistors, you can add a small resistor in series with D3, but this will increase the nominal class A current, so DO NOT add this R without first increasing the emitter degeneration value or you are asking for smoke..
To dial in the output stage class A current, you can measure it by looking at the DC voltage drop across the emitter resistors, but the real Goldilocks test for too hot, too cold, or just right class A current is to look at HF crossover distortion. If you don't have a distortion analyzer handy you can see severe crossover distortion with a scope. Using a clean sine wave source at 20kHz look closely at the transition region between + to - voltage and back again. This will be most visible for modest output voltages driving a healthy load (600 Ohm?). With a distortion analyzer I set the class A current so the crossover distortion components were well below the other distortion products. With just a scope, the best you can do is see it go away, and maybe add a little margin.
I would be tempted to use all transistors and diodes from the same batch (so they will be from the same production lots and similar), and dial in the operating point, by first starving the output stage until the crossover distortion is visible (or measurable), then increase that class A current until it cleans up nicely. If you can't clean up the output stage "and" still have thermal stability, it's time to tweak the design. Note: You need to tweak the bias so it is clean at minimum rated supply (+/-12V), while still thermally stable at max supply (+/-20v?). The nominal current in D3 increases with this rail voltage so if it is clean at +/-12v it will will be cleaner (and hotter) at +/-20v.
I suspect this design and values were dialed in for slightly different parts and different layout.. In such designs the details can matter.
Good luck and have fun.
JR