Ok, I tried to upload my game to see how it runs on the page and... It turns out that some features are 'insecure'. For example, I can not use the Cache storage that is used by my game to cache downloaded sprites for future use. So, the question is - are there any options to use those modern JS features?

You can try enabling Shared Array Buffer support, it will add some headers that may allow you to access more browser APIs: https://itch.io/t/2025776/experimental-sharedarraybuffer-support

Unfortunately - no luck...