Despite the ECHR not being an EU institution, the optics of a European court having jurisdiction over the UK should not be underestimated. This YouGov poll from 2014, shows that the British public favoured withdrawal from the ECHR - the full data shows that the convention is especially unpopular with Conservative and UKIP voters - and this is before the Brexit debate entered the ring.
The approach described by Barnier is consistent with the current policy of the UK Government in their negotiating position outlined in the document The Future Relationship with the EU published in February. Within this, under "Part 2 - Other Agreements", point 31 states:
Cooperation will be underpinned by the importance attached by the UK
and the EU to safeguarding human rights, the rule of law and high
standards of data protection. The agreement should not specify how the
UK or the EU Member States should protect and enforce human rights and
the rule of law within their own autonomous legal systems.
This reinforces the UK Government's position that the EU should respect the UK's independence and autonomy in all matters, including on the subject of human rights - and vice versa. Indeed, the Government reiterated this at the beginning of February:
A Downing Street source said: "We are fully independent and our
approach to a free trade deal will not be bound by our previous
"Nor will we agree to obligations which the EU has not required of
other countries which it has signed comparable free trade deals with."
This is a reference to the existing Canada-EU & proposed Australia-EU free trade deals, among others. Obviously neither Canada nor Australia sign up to the ECHR, due to not being members of the Council of Europe, and therefore the UK doesn't agree that a free trade deal should be contingent on this.
Addressing business leaders and diplomats in London, Mr Johnson will
reportedly call for the UK to be treated as an "equal" in the talks
and demand "no alignment, no jurisdiction of the European courts, and
no concessions" with Brussels.
This point is key to understanding the Government's negotiating position. The Conservatives have just won their largest election victory since 1987, partially on the back of their whole-hearted opposal to the jurisdiction of European courts over the UK. To avoid being accused of a "Brexit in name only" negotiating position, and the possible resurgence of the Brexit party, it is important to be seen to be resolute and uncompromising on this point.
As to what the decision will allow the UK to pursue, the previous Prime Minister, Theresa May, had been a key proponent of a "British Bill of Rights" since she served as Home Secretary in the Cameron administration. After becoming Prime Minister in the wake of the EU referendum, it was widely reported that a side-goal to Brexit was to remove the UK from the ECHR for the proposed British Bill of Rights to be implemented and to have teeth.
As the ECHR enforces minimum standards on members, the only practical reason to repeal it is to reduce these minimum standards. In particular, the Government recently lost a case regarding the police retention of DNA.
Current Cabinet members also have misgivings about the ECHR. Priti Patel (Home Secretary) & Dominic Raab (Foreign Secretary) have both warned that "the ECHR has been repeatedly "abused" by European judges". Outlining his opposal to the ECHR, Dominic Cummings, the Prime Minister's Chief Advisor wrote on his blog in 2018 that "The ECHR creates [...] legal problems all the time", going on to say:
If I get involved in politics again, then a referendum on the ECHR
should be high on the agenda — and bear in mind most people probably
think we’re already leaving it because of the 2016 referendum, so
imagine how mad they’ll be when they realise we’re still in it.
It seems now that Cummings will get his wish without a referendum.
In conclusion, then, the motivation is a mixture of electoral and practical reasons. The Government wants to be seen to be robust in their negotiating approach, and their stance against the jurisdiction of international bodies - especially European courts - over the UK, to please voters, while also allowing the Government to implement stricter laws on certain areas that it cannot currently due to protections enforced by the convention.