Reasons for Popularity of PowerPC for Embedded Designs
To get the price down to where the gaming consoles require, they all needed custom chips that included a CPU (or three). Of course Microsoft switched from an Intel CPU to a PowerPC, because there was/is no way that Intel will allow their CPU in a custom chip-- especially if it wasn't Intel that was fabbing the chip.
At the time the XBox 360 was created, the PowerPC was the fastest and most reasonable CPU to use. This is no longer the case, where ARM has beat it out. I predict that ARM will be the CPU of choice for the new round of gaming consoles that should be out in the next year or two.
While there are new PPC devices, there are also new 8051 and Coldfire devices. So this, by itself, is not a good indication of how "current" the PPC is. New ARM devices outnumber new PPC devices by maybe 50 to 1.
Now to directly address your questions:
Is it the high scalability and application programming interface compatibility within the PowerPC family, from small microcontrollers to processors for high performance computing?
The PPC does not currently offer any scalability advantages. The ARM is actually easier in this department since that CPU was designed with multi-core processing in mind.
The PPC does not offer any API compatibility that ARM or other CPU's do not also offer. Modern software is written completely in a high level language, and so the CPU architecture does not play into API compatibility. Almost nothing is written in assembly language these days, especially on high performance 32/64 bit CPU's.
The robustness of the design and experience with certification of functional safety?
It is unclear on what you mean by this. For most embedded applications that do not require life-safety, military, or aerospace levels of reliability, the PPC offers no advantage today. ARM's have been proven out just as much, or even more, than PPC. For life-safety, military, or aerospace then there might be an advantage but those markets tend to lag the rest of the world by several generations anyway.
Support of high temperature ranges, especially fan-less designs for high temperatures?
The ARM is a much lower power architecture, which is why ARM is used in mobile devices while PPC is not. Lower power = lower heat = much easier to handle high temperature ranges. Advantage ARM.
Backward compatibility (support of legacy software) is one driver, of course. But there have to be more.
Why does there have to be more? I'm positive that this is why 90% of current PPC designs are still using PPC. The other 10% is because some people are just stuck in their ways. There are many examples of old architectures that continue to be used for no good reason. You can still fine Z80 and 6502's being put into new designs, and nobody is calling those good or currently popular.
The reason for the PPC popularity is that it was the right CPU at the right time in the market. Before that it was the MIPS CPU's. Now it's ARM. You still see PPC used because some things just take a long time to die out. There are still MIPS designs out there too.
@NichHalden was also completely correct on this subject.
It was offered cheaper than any product from Intel or AMD when you don't need latest generation CPU's.
e.g. Xbox and PS3 didn't need the fastest CPU around, they just need a decent amount of multi-processing. They also needed a long term delivery program with the price declining to ensure long term viability and price of the consoles.
Xbox and PS3 where originally sold at a loss to start the ecosystem, and then over time they are cheaper to produce. The components don't fundamentally change but do get significantly cheaper so to ensure a healthy margin in the boxes, unlike desktop PC's which are expected to follow the latest CPU generations.