I think the key phrase is "a British bill of rights will strengthen the hand of the judiciary over parliament." That is, one branch of the ruling elite (politicians) are afraid they will lose some of their power to another branch of the ruling elite (judges). They fear the judges messing about with their new legislation when somebody appeals to the High Court or Court of Appeal about it being misapplied or misinterpreted. This has nothing to do with supposed public opinion.
The only serious hostility towards the Human Rights Act comes from the print media such as The Daily Mail, Daily Express, The Sun and (more subtly) The Times. These publications cannot tolerate anything remotely life-affirming and have a sustained campaign going to undermine the Human Rights Act by printing lies, misrepresentations, and partial truths. Unfortunately, people do tend to believe what they read. And "the public" tends not to notice the Human Rights Act, except when an individual has to assert their rights under it. That person then appreciates how invaluable it is. But relatively few people have this kind of experience, or are politically aware enough to appreciate its value.
I think it is not cynical to assume that, if the politicians want to tinker with the Human Rights Act, it's not going to be really about expending its provisions, but to give them another tool to control and coerce people. What kind of "responsibilities" are they talking about, and more important, who would you owe them to - the Government?? Eff right off. Show me a government in the past 150 years that got voted in by a majority of the people who actually voted, much less by a majority of the whole electorate. In other words, show me a government that's ever had any legitimacy.
LIberty is starting a campaign to raise awareness of the Human Rights Act and to kill off the myths (lies) that have been spread about it. Granted, Liberty has its own shortcomings as an organisation. But at least they're doing something. Is any other organisation or group?