Minecraft – How to protect private items in chests in vanilla Minecraft 1.12

Unfortunately, it appears the answer is "you can't, or at least not without spending a prohibitive amount of time on the investigation, and even then you won't be sure of anything."

If someone finds a solution that's less time-intensive, I'll happily accept it instead of my own bleak conclusion. But after a lot of research I suspect there isn't one. I've updated the question to clarify that solutions ideally won't require hours of tedious manual work or monitoring.

I appreciate the answers that have been submitted so far. Unfortunately, manually searching everywhere for a stolen item would be too time consuming, and even upon finding it you couldn't be sure who put it there; it would be easy for a thief to frame someone else. And watching a potential crime spot continually until the crime happens would take too long since there are several possible griefers, many locations that could be griefed, and thefts are often spread out in time.

So for the time being the answer seems to be:

1. Use the techniques that I outlined in my question, whether using InvEdit to inspect player inventories, or else manually copying their [username].dat files to inspect inventories and Ender chests. If it's in their inventory or Ender chest, there's no question who took it.
2. Or give up and install Bukkit or Spigot and an anti-griefing plugin like WorldGuard or Prism.

We switched hosts and installed Spigot and Prism, and it was 100% worth the effort. Spigot runs even faster than Vanilla Minecraft did, and Prism is awesome for tracking down offenders and reverting their evil deeds.

Due to the fact that I was only testing this system in creative and that creative inventories work differently than survival inventories, this bug occured, in survival however, it doesn't. In a creative inventory, a player saying to the server that he has a certain item, gives the player that item.

So if the client thinks it has a certain item and clicks that item, it will tell the server that he has the item, thus creating the item. In a survival inventory though, the server won't accept the player just saying they have a certain item, so it just discards the ghost item. leaving the client only with the real item.