Speaking as a designer, not someone troubleshooting specific power issues, I wonder if DSP firmware can be designed for a more gradual power up.
The ARM Cortex-M chips that I’ve developed on have individual power enables for each internal peripheral. Of course, it’s common to immediately turn on every peripheral interface that’s needed, but it seems like it would be possible to have a time delay in the firmware startup such that the inrush current would not be so drastic.
I guess the real question is: How much of a problem are DSP modules and initial power draws? If it is a significant problem, then it might be worth pushing manufacturers to update their firmware. We humans won’t notice a few extra milliseconds - even hundreds of milliseconds - at startup, but the power supply would!