@leafo thank you very much!
This explanation showed that there are no restrictions on file types or how we load them that prevent HTML from loading. So I rigged a little deeper and it turned out that Phaser asset loader allows different path specifications when the game is hosted in domain's main folder, but more "strict" in terms of how it finds assets if the game is inside some other folder.
That's why the game worked fine locally on localhost and on it's website, but on itch.io where path is like "https://html-classic.itch.zone/html/10661524/dist/assets..." it failed.
I will change the code to make it work. Hope it can be useful for other people who might run into similar problem.