" In more analog mindset, does it make as much sense to evaluate a separate music server in the same terms as one might consider a cartridge to a turntable?"
Music server hardware, no, not much if at all anymore really in practice. Music server software that runs on the hardware, yes.
A music server is a computer program that streams data from files. It only has to work reliably as designed like any other computer server for any other app must. It need have nothing specific to do with the making of sound other than supplying the data which any properly operating computer device can do just like always. What is in the files themselves will have a lot to do with sound....that is where the source quality is determined, much like the source quality of a record or tape.
In some cases, some music server programs may provide features that change the data it gets from the file before sending it and hence the sound may be changed in some pre-determined way, like a loudness leveling feature for example, to help make all files achieve similar loudness levels despite differences in how they were originally mastered. . But that is part of the server software....it can run on any supported hardware and do its thing just fine. Anything is possible there but as a result of the specific server software running not the hardware. Same true with the streamer software. Both server or streamer software programs may have features that allow it to change the sound in some predetermined way.
Music making starts once the streamer starts sending the data it received from the server or any other streaming source to the DAC. Now the digital data must be converted to analog accurately in real time. Here is where the music making starts, similar to a cart on a turntable and where ones traditional concern for good sound reproduction from the source is better served.
Bottom line: pay attention to specs and features of music server software. Choose the ones you like that provide the features you need or want. With some you may have the option of running it on different supported hardware platforms. With others, the vendor will decide for you what hardware to run their server software on and both come bundled together as is the case with many music servers marketed to traditional home audio users. Expect to pay a premium for plug and play music server devices especially if marketed to audiophiles and pay attention to the specs (like resolutions and formats supported) and any features that might help distinguish one server from another.