Well, I worked it out. I did end up buying a Pi2 and a Banana Pi. The Lubuntu image fort he BPi worked great, looks great, and I set up a Samba share with no problem. The Raspberry Pi was a little more difficult, though to be fair it needed sound over HDMI to be setup, the BPi has no use for sound so I didn't even test it.
I tried a few different distros but couldn't (and still can't) get sound working across all applications on the Pi. Mplayer won't play sound, VLC will but won't play video. Some audio players worked, some didn't. I tried all the tricks: uninstalling Pulse, forcing audio through HDMI instead of the headphone jack, etc. No dice.
Ultimately I tried OMXPlayer, which is pretty cool. I wish the timeline seeking were a little more fine-grained but it's fine. The playback is really smooth and it looks great, better than any other player I got running. OMXPlayer was written for the Pi (I think) and it shows. The only issue is OMXPlayer is command-line only, so I added a desktop entry for OMXPlayer so it would play the file with a double-click, no typing required. The desktop entry I used was copied from this thread.
Other hitches not unique to the Pi: I forget over and over how big of a dick Ubuntu can be with external drives. If it's a thumb drive or SD card it usually allows any user to write to it, but a large hard drive is locked down and only root can write to it. Chmod and Chown don't work as they should, and strange workarounds always seem to be needed. This was my experience when setting up my old laptop as a file server too.
Also, Linux audio still seems to be where MS-DOS was at its height. I didn't have to mess with channels or IRQs or whatever but I might as well have. Not to mention the confusing overlap of interfaces. Do ALSA and Pulseaudio do the same things? Pulseaudio is the newer one and was supposed to replace ALSA, yeah? I don't really know, I seem to remember it being explained like that to me at one time. Audio and wifi were always the biggest issues for me to get working with Linux a decade ago (and video acceleration, but in that case if you didn't have it you probably never would so no tweaking was needed either way). I still remember the days when updating Ubuntu meant keeping my fingers crossed because wifi and audio were very likely to have been broken in the process.
And don't get me started on the problem of new versions of certain programs (or even the system itself) keeping old config files in their old directories but ignoring them, and you're supposed to know this somehow and seek out the new place to create or edit an identical config---once you realize the program has stopped working right, of course. Some programs have three or four copies of the same config file scattered around and it's a fun game of elimination seeing which file actually affects the program's behavior.
Wait, was I ranting on Linux or was I talking about my home media setup? Anyway, it works well, I'm very pleased. The Banana Pi isn't set up for transcoding, not that I think it could do anything heavy anyway, but with the Raspberry Pi I don't think I'll need any. It does worry me that I only have one working video player on the Pi, in my experience every video player has a few files it just can't play properly even though every other player can. Still, I think I'll be fine.
Now to start copying all the videos on my watch list over to my network drive.