This seems to have become a bit of a myth - that when you apply DSP over the whole range, you lose top end sparkle. Not so in my experience.
Firstly, if you are using DSP as the basis for the crossover, then there's no escaping the fact that DSP is being applied over the whole range.
Secondly, if what is being referred to is applying EQ across the whole range, then that is certainly a choice - and one of the choices that can be made is to enhance the top end sparkle by de-emphasising the rest of the range. In the more sophisticated approaches to DSP such as Dirac, this is formalised by allowing the user to define a "target curve" that tells the DSP engine what the desired FR profile should look like - and its shape is under the user's control.
Certainly, with powerful tools like DSP, there is enormous scope to screw things up if you don't know what you are doing, but there is also enormous scope to create a FR profile that actually matches your listening requirements.
If you are using an amp with DSP room correction built in (eg Dirac in NAD or RoomPerfect in Lyngdorf) the full range signal from the preamp either bypasses the DSP (filter off) or is subjected to it (filter selected). This happens to all frequencies even if the version of Dirac (as mine is) can only adjust sub 500 Hz frequencies.
I have Dirac in my NAD M33 and I've used RP in a demo Lyngdorf 3400. When the signal is subjected to a filer, there is no doubt whatsoever that the top end sparkle is compromised. This can be easily demonstrated with speakers that offer an abundance of sparkle! In less good speakers, this loss of detail may not be noticeable.
There is no choice about this - I’m not saying that DSP is all bad, but in full-range amps you cannot prevent the higher frequencies from this DSP circuit. Unless, of course you split the incoming full range signal into bass and the rest before DSP and amplification and DSP is applied only in the bass amp . Work it out, this is the only way to protect high frequencies from DSP, but of course bi- or tri-amping is required to achieve this.
DSP can offer an apparent improvement in sound quality but it should (as others have said) be used as a last resort. There are plenty of other steps to be taken that should negate the need for DSP, but some systems (mostly multi-speaker AV ones) are complicated and the easy route is to bung DSP at the problem.