If I recall correctly, Apple tried to do that and it (rightly) elicited howls of outrage. What you're asking for is for people's own computers to spy on them on behalf of the authorities. It's like having people install CCTV cameras their own homes so the police can make sure they're not doing anything illegal. It's literally Big Brother stuff. Maybe it would only be used for sympathetic purposes at first, but once the infrastructure is built, it would be a tempting thing for the authorities to abuse (or just use for goals that are not universally accepted, like banning all pornography).
I don't want my browser to report me if I encounter illegal materials. I want the browser to anonymously report the website where they are, at most and even that, only if I don't disable reporting.
People do install cctv cameras in their homes but they are (or at least believe to be) in control of what happens with the footage.