JavaScript (TypeScript) has access to cookies (and thus JWT). This should be handled by web browser, not JS. In case of log-in, in HTTPS POST request and in case of response of successful log-in, in HTTPS POST response. Then, in case of requesting web page, again, it should be handled in HTTPS GET request. This is lack of using least permissions as possible, JS should not have access to cookies.

JavaScript needs access to the cookies, they are the data storage for a given site.
To protect them, the browser silos them to the individual site that created them, that's why developers haven't been able to easily load cross domain content for years, to mitigate XSS attacks.
The security relies on the premise that the only valid source of script is the originating domain.
The flaw here was allowing clients to add arbitrary script that was displayed to others.
You're dead right that only the way to fix this is to do away with JavaScript access to certain things, but it will require a complete refactor of how cookies work.
I haven't done any web dev in a few years, this might even be a solved problem by now and we are just seeing an old school implementation. 🤷