When I get the time I will visit the Royal Library in Stockholm, where they keep all books ever published in swedish. They got to have the full version. Thanks to you, Fwiffo, I know where to look in the book. It's pretty boring to read - at least for the second time. :-)
I have read Linneus Laponian journey in swedish. The original text is in swedish with lots of latin expressions. The notes were meant for a diary and not for publishing. However I found no mentioning of Tablut. Now, the version(s) I looked upon were incomplete, but there was a description of another game where you throw sticks.
When I bought a board with pieces made in Sweden a few years ago, the included rules stated that white wins if the King reaches a corner square. That rule should make the win ratio more equal, I suppose.
I will try to find the swedish original and see what it says about the rules. I didn´t know this was the only detailed source to this truly fascinating game. Since most information seems to be second handed, I will see if this brings any clarification.