I now believe it to be an issue with saddles specifically, with vanishing saddled horses being an unfortunate and sad side effect.
Not consistent, but so far 1 out of about 8 scheduled shutdowns/restarts have seen saddles and saddled horses completely removed. I have only been playing on a single world running on a 1.14.6 Minecraft server on Ubuntu 19.04. Scheduled restarts (and backups) twice daily. No mods. The backup that was done at the time of the restart also has no horse/saddle, and my installation backs up on start, so I think it is when the server shuts down that the saddles are deleted
Originally I thought this was a horse issue, but now I have determined that non-saddled (but tamed) horses never go missing. I'll leave my horse with a saddle either leashed to a post or free in a pen/enclosure. I'll log out, and when I come back, the horse will be gone. No nearby dropped items, either, implying it wasn't killed.
I exported the world and opened it up with MCC Toolchest PE and compare the world missing the horse with a world that still had the horse and found it has been entirely removed, and no saddles in any of my chests (I know I had at least 2). I'll note I did not use that utility, or edit/open the game files in any way the first 2 times this happened.
Another user here has reported this I think, likely the same issue, but evidently not using a server I think.