A DSP on sound cards was more common before modern CPUs which have more processing power and DSP like functionality, especially SIMDs. There still is the possibility of off-loading the CPU but most audio processing only uses a few percent of a modern medium to high performance CPU's processing power. An SIMD often has more performance than a DSP albeit at a higher cost. Perhaps the PS4 is trying to save electricity as consoles need to be more power efficient and a DSP wins over an SIMD here. However, the PS4 likely has multiple SIMD units which could do more complex audio processing faster and the overall hardware cost could be reduced by a small amount. It would be interesting to hear why the PS4 engineers included the DSP.
As a guy who has done nothing but audio coding my whole career I can agree that being able to move everything off of DSPs and on to the high performance main CPUs that came along is a great thing, at least for us writing audio code. Auxiliary processors can box you in and are generally just a PITA to deal with. They are also often closed off, only presenting some high level API to interact with.
I might be cautious in saying the PS4 has a "DSP". It is not targeted for general purpose audio processing, the main CPU is what does all that. The "DSP" mostly deals with encoding and decoding. Is that a DSP, or is it just another auxiliary processor that is cost effective at that task? There might be some other DSPs on the machine I am unaware of, there are a number of them that perform various tasks, but none that synthesize the game audio like some might think.
I'm also not sure if you'd say the Cell is super DSP-like. Its instruction set is quite general purpose, but designed for modern SIMD computing (lean and mean). Its memory architecture is not at all general purpose, and more DSP like, I guess. It's interesting, to say the least.
Reading this thread I now kind of wish I had a Falcon way back when it came out. It would have been a great machine to code the audio software I wanted to develop with its 16-bit capability, and back then the DSP was needed with the main CPU being too weak for anything interesting with audio. I guess I had moved on to PCs and sound cards by that time though.
Relating this back to the Amiga I wouldn't actually mind if someone came out with a new sound card for it that had a bitchin' DSP-like processor on it. Given the Amiga tops out at 68060 you kind of need it if you want to do lots of crazy stuff. Either that or some of these new accelerators that could move us way past 060 performance, that would be even better.