Oh shit you're right, it does do that!
For others, here's what I did:
- Open Kate
- Click the "Sessions" menu, then "Save session". Give it a name.
- Close Kate.
- Open Kate, it prompts for which session you want. Click to open the session you just saved, but also click "Do not ask again".
- Open a new tab. Type something. Open another, type something else. Don't even save, just close Kate.
- Open it again, everything is where you left it!