Every technology has its benefits and downsides. As for me, I much prefer DSP over passive filters, and I use to design passive filters as an occupation. The problem is not so much with capacitors, but with inductors.
If you want low DC resistance, which is important to keep the damping factor high for woofers, typically inductors with ferromagnetic cores are used, which introduce hysterises and accompanying power losses. If laminated steel inductor cores are used, add to that power losses due to eddy currents in the steel laminations. The power losses increase with frequency and are especially audible in the upper midrange. (That probably is why some of the better 3-way speaker designs will use cored inductors on the woofer only, but stick with air core inductors for the midrange and tweeter).
Air core inductors eliminate the core power loss issue, but typically have higher DC resistance, thus reducing amplifier control over the woofer. If you have a tube amplifier with output transformers, you already have high output impedance so it probably is not going to hurt it much more. But, if you have a low output impedance amplifier, adding high DC resistance in series with your woofers negates much of the benefit of having low output impedance.
Both of my primary stereo systems, one in my family room and one in my office, are all active. Indeed, I removed the passive crossovers from my family room speakers and modified them to be all active using DSP, along with additional bracing and damping material. Objectively they measure better. Subjectively they sound better, especially in the bass, though I did not do a side by side comparison since I modified both speakers at the same time.
Finally, all filters indroduce phasing issues. With passive filters, though, the phase response of the filters is affected by the complex impedance of the drivers. With active filters, although the drivers will introduce their own phase characteristics, the phase responses of the filters are textbook. Combine that benefit with the availability of adding time delay to drivers for the purpose of phase alignment, and that provides additional benefit for going all active.