WooHoo! I did something useful in finding a bug! ;-)
As for the highscore stuff, I created a simple (and I stress simple) "online leaderboard" solution for my game jam entries a year or so ago (based off a blog post from another random game jammer). I've used it for a handful of my jam games (all Unity webGL builds) without issue. I'm not sure if it'll help you track down your particular issue, but, for what it's worth the two main gotchas that I had to overcome to get the leaderboard working in webGL builds were:
- My leaderboard needed to be hosted on a SSL secured site. Without a valid HTTPS connection, the webGL build wouldn't trust the connection
- I had to add the itch.io domain to a cross-domain whitelist in my leaderboard website code. I know next to nothing about Web dev, so I was blindly following the blog post I was reading, but, from what I recall, since there's cross-domain shenanigans going on between itch and my web host, I needed to specifically allow itch.io to play in my world
By my recollection, neither of those items were issues if I was building Windows or Android games. They were only issues in the webGL world.