It also had not occurred to me to ask, but are you reading your Gmail using the Gmail web interface or an e-mail client? If the latter, what client and are you using POP or IMAP access to download messages?
The only thing I can tell you as far as the Gmail web interface is you could toggle the "Show/Hide" for the "Send & Archive" button in General settings as suggested in the Gmail support thread previously referenced.
You might also want to check if some bizarre filter was created that causes message archiving in the Filters and Blocked Addresses setting.
Otherwise, if you're using a client, something could be filtering messages that causes the IMAP server to archive them. If that's the case then you should find all of your disappearing messages in "All Mail".
The reasons and ways messages can start to disappear are not necessarily simple. Please also have a quick look at the following for ideas of what you may dig into as part of this:
I realize how frustrating this is but all I can really do is point you to references regarding what to check since it could be any one or combination of factors.
Please report back on your progress (or lack thereof) and if you find a solution that applied to your situation please share it so that it goes on the "to be checked" list for others who experience the same problem.
Brian AKA Bri the Tech Guy (website in my user profile) - Windows 10 Home, 64-Bit, Version 1803, Build 17134
In a modern society where everyone thinks their opinion deserves to be heard nothing annoys me more than individuals who mistake their personal preferences for fact.
~ Commenter TheCruyffGurn on the The Guardian website, 8/13/2014