Angle brackets with Finnish keyboard

I have a Finnish PC keyboard connected to this Mac (macOS 10.12.6), and besides the quirks of Mac OS X wanting to force a Mac keyboard emulation, it worked mostly fine. The angle brackets (ie. <>) came from the key at the right of the left shift key, and the tilde symbol (~) came from the upper key at the left of the return key.

Then I was trying with some new mice, and OS X became completely confused about a new keyboard having been connected, and asked for the keyboard type. I don't know what happened, but it completely broke the keyboard layout. I think I might have accidentally chosen the US keyboard layout or something. I tried to change the layout back, but nothing worked. The "change keyboard type..." button in the keyboard system settings was non-functional (it could be clicked, but nothing happened). The layout was completely broken, and I couldn't fix it in any way.

I found a tip online that deleting the file /Library/Preferences/com.apple.keyboardtype.plist and restarting would make the system ask for the keyboard type again. It worked, and afterwards the angle brackets came from the proper key... but the tilde key was broken. Rather than giving a tilde, it gave a completely different key code. It became, in fact, completely impossible for me to type the tilde character in most software.

I did the trick again, and that restored the tilde key. It's working as it should. However, now the <> key and the §° have swapped places. And there seems to be no way of swapping them back. Needless to say, having <> in the wrong location is highly, highly annoying and cumbersome especially when programming.

In the "Input Sources" tab of the system keyboard settings, the < key is shown as being below the esc key, in the Finnish layout. According to who??? It has never, ever been there, anywhere, at any point. Just look at the Finnish Mac keyboard (make a google image search if needed). It's on the right of the left shift key, as it is in essentially every single keyboard layout of any language.

Deleting com.apple.keyboardtype.plist isn't helping. How do I fix this? And don't tell me that the only way is to install some kind of third-party key remapper.