The question isn't "should or shouldn't". The question is
how.
Web standards are supposed to be based on a) consensus and b) at least two independent implementations.
As an example, let's take a recent Chrome-only non-standard that made HN frontpage, WebTransport: https://web.dev/webtransport/
The status of this API? "Editor's Draft". That is, scribbled on a napkin [1]
--- start quote ---
These drafts have not received formal review and are not endorsed W3C.
These drafts MUST NOT be cited as W3C standards and may or may not become W3C standards.
--- end quote ---
Guess what its status is in Chrome? Already shipped, and advertised as available.
As with other such non-standards expect HN comments decrying other browsers as being "the new IE" because they are in no rush to develop this.
[1] https://www.w3.org/standards/types#ED