As bonzo mentioned, it's all in the execution.
Simple concept crossovers through experimentation like many did decades ago is a crap shoot if you are really after smooth, predictable behavior. It's a few hundred dollars these days to have accurate measurements and complete electrical models of a speaker. Fortunately this has become much more common in designs coming to market over the past ~5 years. The issue with passive crossovers on horns is that the impedance of a horn is much more complex than a simple dome and especially compared to a ribbon. Each peak in the impedance interacts with any component in the crossover, often well above or below the intended crossover, and by design there are many more peaks in a horn's impedance vs frequency. While there are cases where horns are naturally very flat, most have significant undulations or tilt to the response which needs to be adjusted for the in the crossover. Careful measurement, modeling, and design can yield great results from a passive crossover, but are far from a beginner project. One of the biggest hurdles for the passive crossover is dealing with any physical/acoustic offset in group delay (they delay in time from when a given frequency emits from the horn vs when the signal goes in). Any changes in physical distance from the listener between devices will affect the crossover region, where the physical depth of horns makes the pathways open the possibility of physical offsets being a few orders of magnitude larger than that of a dome and cone design.
Obviously DSP adds components in the chain. If done before a digital source becomes analog, we could make some strong arguments for being less intrusive than any passive crossover with the caveat of being optimal for digital sources, and requiring an A-D conversion of analog sources. Obviously there are plenty who have no interested in that conversion. If it is an option, DSP offers a few very significant advantages when implementing large, multi-way horn systems. First is the freedom to manipulate and filter the frequency response entirely independent of the loudspeaker's impedance. With the tendency for horns to have more irregularities in raw response, DSP allows for exactly consistent filtering by eliminating crossover value tolerances and remaining isolated from speaker impedance variations. This is particularly useful for any narrow or sharper (=higher Q) adjustments to the response. Narrow or sharp filters are tricky to hold tight tolerances with in passive crossovers as you have multiple parts interacting, each with their own tolerances. While it is certainly possible to match significant sensitivity differences between components in a passive crossover, it is never a simple 1-2 part adjustment, as most any passive component change will impact the frequency response and behavior of crossovers and the like. With an active or DSP implementation, relative level offsets are trivial and precise. Of course the obvious and most significant advantage of DSP implementations come in the form of delay adjustments which are not practical in the same manner passively. This frees us to only have to consider the point of where the speaker radiates sound and where the sound sprays without trying to account for group delay from horn pathways or filtering. Where physically any offset will make changes in multiple dimensions, in DSP a few clicks of the mouse spares us having to re-construct or build in excessive adjustment features.
Both approaches have merit depending on someones perspective, use, preferences, and let's face it... ideology. As always, implementation always out performs technology poorly implemented. The market interest in big horns makes things a bit tricky as well, since a large portion are curious about horns and the options they can open with low powered tube amplifiers. Those fond of such tube options are more likely to be fans of analog sources and less quick to incorporate a DSP solution. While that may be a tendency, it is certainly not the rule, as many others mix tubes into the output of their digital music library to get the sound they are after. It might prove entertaining to juxtapose a digital source feeding a DSP crossover with high quality DAC's driving say a six-pack of triode monoblocks.