I just realized that I've been playing the game this whole time without the cinematics. There aren't many of them, and they aren't crucial to the gameplay, but it's nice to have them.

Apparently the Wine / Wineskin install I made won't play the cinematics without a bit of tweaking.

I followed this guy's instructions here, even though it's a 3+ years old posting:
http://appdb.winehq.org/objectManager.php?sClass=version&iId=15502&iTestingId=36817&bShowAll=true

The version of Wine I used in my Wineskin wrapper is 1.1.33 (older, yes, but nearer to the ones in the instructions link above). Then I made sure to get those dll's mentioned in the instructions, regvr32 them, and set them in WineCfg as "Native" (NOT "Native then builtin").

EDIT: I just confirmed that you can use one of the latest versions of Wine. The important part is just to grab those native dlls and tell Wine to use them as native.

After doing that, i got the cinematics working again.

Hope this helps someone else!