So it's universally acknowledged and there can be no argument that Google Hangouts, though having some nice features, is pretty bad when it comes to actual chats. I mean really bad.
Want to look at a chat you had 6 months ago? Good luck finding it. (I'm not even sure that functionality is possibly any longer, which says something. Even if it is still around, the fact that it's so obscure that the incredibly tech savvy can't find it readily is an abject failure.) Try finding that old chat with the gmail app? Nah, not happening. Even when you could, it was HORRIBLY broken.
As for missing chats from people, Hangouts is actually worse. You see, if you have Hangouts logged in on one of your devices (Phone or Tablet), the message will be received. But it won't TELL you that you've got the message if you log into your desktop. If you are lucky the message will appear as non-new (already read) message in your inbox. However there are times when it won't even show up there.
So how do you know if someone sent you a message? Well, if you notice that you've received something, then you can go dig through your "Chats" section on your desktop browser with gmail.com.
So yeah, Hangouts is arguably the most broken mainstream app from Google to date, which is really sad. Not to mention the bitch is a pain to log out of. You have to do five clicks and a couple of scrolls. Bad. Bad bad bad.