In the context in the post, I’m saying that before we get rid of an assumption of probabilities, it is easier to first get rid of the assumption that we have a real-valued L (proving it from other assumptions instead). We do this by applying VNM, still assuming mixed strategies are described by probabilities. I finally get rid of that assumption a few paragraphs later. I edited the post a little to try and clarify.
VNM assumes you have preferences over lotteries, which imply belief in frequencies/probabilities.
In the context in the post, I’m saying that before we get rid of an assumption of probabilities, it is easier to first get rid of the assumption that we have a real-valued L (proving it from other assumptions instead). We do this by applying VNM, still assuming mixed strategies are described by probabilities. I finally get rid of that assumption a few paragraphs later. I edited the post a little to try and clarify.