I'd say that it's the other way around, and native elements that are not system-dependent (say, a video player) should lose prominence. There's no reason to have a "native-looking" button on a web page, which otherwise cannot (and should not) be made native-looking.
Instead I expect the industry to stabilize around a few widespread sets of web components, much like a lot of CSS for controls stabilized around Bootstrap's styles.
Which controls stabilized around Bootstrap styles (now there's a a name I haven't heard in a long time)?
No, industry should not and will not stabilize around "few widespread sets". For the simple reason: it's extremely difficult to create a proper userland UI control in the browser. How many custom drop downs fail even the most basic keyboard behaviours? How many custom UI elements fail even the most basic screenreader interactions? etc. etc.
What you really truly need is a rich browser-native set if controls, and https://open-ui.org/ is slowly working towards that.
Instead I expect the industry to stabilize around a few widespread sets of web components, much like a lot of CSS for controls stabilized around Bootstrap's styles.