I love DSP. It is used in all professional audio applications. Any concert you go to is using massive amounts of DSP in several different ways to make the system sound better. My home speaker builds sound better with DSP. I use mini DSP's for it. Yes, this usually adds an extra conversion but when used properly it can compensate for room and speaker issues. Examples: 1) I delayed the tweeter on my MTM living room speakers. The delay is .31 milliseconds which I'm guessing is about 3-4". When measuring with an RTA mic on the program Smaart (which is what 99% of live sound engineers use) you can see the phase problems between the tweeter and the mid/bass drivers go away. They line up perfectly. With DSP and the bi-amping of these speakers I could try an infinite number of crossover points, frequencies, slopes, etc. and then find which sounds the best. Then I can use a little bit of EQ to get rid of any problem frequencies caused by the room and my speakers have almost a flat frequency response in the area on my couch where we sit and listen.
Yes, there are disadvantages. Danny from GR feels that the miniDSP can "take the life out of the sound" and I understand that DSP is another digital conversation (two actually) and this does change the sound, but if you're already starting with digital from your TV, steamer, computer or bluetooth then I don't think another conversion does anywhere near as much harm as it helps. But it has to be used correctly. The room analyzers that do it automatically are basically doing what live sound engineers do. Our advantage is that we use these things as tools and also use our ears to keep it sounding natural.
One more point, if I may. I am working on building some speakers with an analog crossover for a fully analog system with record player. I don't like how my records sound through my DSP. I agree with Danny from GR that the life is taken out of them. But that means that I need to measure how much delay the tweeter needs and then do that in the speaker build and get the crossover right. I could always implement an analog EQ, but EQ is always adding phase shift so you are altering the sound in some way.
So, I think DSP is better for audio that is already digital. But when dealing with an analog source, it would make sense to keep it all analog, which means really nicely designed speakers and passive crossovers.