If you have Rockefeller on the ticket, you're going to have more than just Civil Rights. But at the same time, Nixon isn't going to win because of the backlash he would have faced from Conservative Republicans- a backlash that may have resulted in a third party candidate running from the right and splitting the vote, giving Kennedy an even bigger victory. Indeed it is easy to forget how hostile Nixon and the 'Rockefeller Liberal' platform was received at the 1960 Republican National Convention- the man was booed and heckled, and even among pro-Nixon liberals and moderates the platform was problematic because it was drawn in the backroom and would have scuttled the Southern Strategy, and indeed overall the programme drawn up between the two men was simply too alienating to key Republican votes in the Midwest and South.
If Nixon can get Rocky onto the ticket (as he did indeed try and do IoTL, only to be turned down by Rocky who made impossible policy promises such as a National Insurance System), then it's unlikely he'll win, be it because of a split of because of how alienated the base would be. Indeed, it's unlikely the Liberal Republicans would even survive the election if they caused a larger defeat. 1960 was their Alamo- they were on the final fight with change that had been happening on for 44 years, a change with so much momentum behind it that it wasn't going to suddenly be stopped by Nixon forcefully pivoting the party to the centre. At his core, Nixon was a conservative, a by-product of his upbringing, and although he was a self described 'liberal' on Civil Rights (which he refused to budge on in committees due to his personal and moral standings on the subject), you aren't going to get a more 'liberal' Republican party from him just going for Rocky.
And you want both parties to be strong in all major regions of the US... but it doesn't work like that? It's never worked like that, and indeed can never work like that, and if Nixon and Rockefeller got in and for unknown reasons could lurch the Republicans back to a liberal toe line, it's hardly going to suddenly make both the Republicans and Democrats strong in all the major regions.