A person runs into this problem in other use scenarios as well. The difficulty boils down to content protection. Cable boxes, satellite boxes, TV’s and Blu-ray disc players don’t usually have multi-channel digital out, other than HDMI, due to HDMI licensing agreements (and most DSP units and digital sound cards for PC don’t have HDMI in). Most often you find analog out, and in some cases digital stereo out. Having said that, my DirecTV satellite box has toslink optical 5.1 Dolby out. So, it’s really important to plan things out according to your exact use case. If you go with bi-amp stereo (and from the sounds of it, that’s what you’d like to do), things are sometimes a bit easier...
You mentioned streamer, and you mentioned AVR, which to me implies home theater. Do you have a streaming music service now, or do you have a particular one in mind? Do you want audio for watching TV to go through this system? The answers to these questions will affect your system choices, because of content restrictions.
For the crossover, you have to choose if you want digital or analog. Analog is possible and exists, but is harder to find, may be more expensive, and may not do as good a job performance wise as a digital crossover. For these reasons, I expect most people wanting to do something like this find a digital crossover to be preferable.
If you decide on a digital crossover, whatever device you want to use will have either a digital input, or a built-in ADC. And there’s the rub. One way or the other, you have to have a suitable digital interface between the source and the DSP, and the options are restricted. An AVR is purpose built for multichannel home theater, so while you might be able to adapt it for your application, there may be alternatives that would be somewhat simpler and save you time, money and space.
A lot of DSP devices are designed specifically for room correction, so they take the range of audio frequencies and divide them into 10, 12, 15 bands or more. What you want to do is much simpler. You want a separate band for each driver in the cabinet, so maybe two or three, likely four at most, for each cabinet. One solution you might consider is something like a Crown 1502. It is a stereo power amp with a built-in digital ADC and crossover, so analog in and analog out, which simplifies connecting it into the system. It would be relatively easy to use for what your looking to do, and would work with anything, at least anything I can think of. That’s especially important if you ever want to reconfigure the system to use different sources, a different preamp, or different speakers. Harman has little charts available online to help explain how their systems work and how to put them together.