He didn't say screen sharing, but "stock share screen", i.e. the interface you get when you tap the "share" button (e.g. when viewing an image) in order to choose which app to use to share it.
Sadly the Google sharing interface, which like many Google interfaces is not the best, is increasingly hard to avoid. F-Stop, which is my most-used Gallery app, does offer a list of apps to use with the "system share" as an option, but that's the exception.
I used to think that the thing that would drive me to leave Android would be some egregious privacy violation by Google. Now I increasingly find myself wondering whether it will be an increasing uniformity and the pushing-out of third party solutions that will do it first (e.g. their "gesture navigation" still doesn't work reliably with 3rd party launchers after 2 years, and they appear satisfied with that situation - at the moment I can work round that because I have other options, but I'm sure they will go away in future).