Again, "yes", the game would have been invented. The Het Kolven influence was very likely only to introduce sticks and balls. There not being too much ice near the coastal areas of Scotland it was only fitting that striking balls with clubs moved to grasslands and dry lands.
"Even the names are similar", said one of you. Bahhhh!
From "Routing the Golf Course", a book every man, woman and child should own...
The Word Golf:
...[W]hile many believe the connection between the words golf and kolf to be compelling, the spellings of golf through medieval times varied widely. It was not uncommon for words to have significant spelling differences, even within the same document. Gowffe, gouffe, and golfe are but a few of the many examples.
In The Story of Golf, From Its Origins to the Present Day (1972), author Tom Scott offers some interesting thoughts on where the Scots may have come up with gouffe. Besides discussing its possible roots in kolf, Scott points out an old Scottish word, howffe, which meant “meeting place.” He makes an excellent case that in a country of varied dialects, such as Scotland, such a progression involving the sounds of “h” and “g,” would not be too unordinary. Golfers, after all, had to have places to meet, and golf links could very well have been named for this big-picture need. Interestingly, Scott does not connect howffe to the Dutch word hof, a word for “courtyard.” Perhaps this would only add fuel to the Dutch argument that golf might have come from their game kolven. It is also quite thought-provoking of Scott to call attention to the Scottish word gulfe, which means “bay.” Could there be a connection between these words? It would have been quite logical for the word golf to have roots in a word that describes land along the coast where golf had become popular.
Golf has so much to do with the physical ground — the land, the course, and the meeting place — that it would make great sense if its name somehow stems from these elements.