Pirelli are required as part of their contract to provide the teams full details and specifications of the tyres for the following season on 1st September the previous year. The teams would have known the details of this seasons tyres by 1/9/12 so would have been able to use this information in the design. Testing would simply tell them how successful they have been.
The issues this weekend have been concerning, and safety is paramount, hence changes should be made, and can be made without the unanimous approval of the teams.
That said, Pirelli adressed the delamination issue as best they could with no unanimous backing from the teams, and this weeks issues seem to me at least to be largely unrelated. The piece by Gary Anderson at turn 4 post the race would indicate that there is more at play at least at this rate, and there are other avenues of investigation which need to be pursued.
Couple that with the teams pushing the boundaries on tyre pressures and cambers, which was also mentioned this weekend, and you have a potential perfect storm.
I am no expert on tyres and high speed racecar dynamics, however, running a tyre at low pressure on a high speed, high-ish downforce track with corners like Beckets, Stowe and Copse is going to cause seriously excessive sidewall movement. Add in a razor edged curb to weaken the tread of the tyre, then you have an issue.
I know that Pirelli have become the popular enemy recently, and this last race has done little to alleviate that, however, they have proposed changes, which were not allowed, they have tried to test, and make improvements, which backfired, and they take the ire of the fans when they are attempting to fulfil a brief given to them.
And they still seem set to continue next season.
If I were in charge of Pirelli, I would suggest F1 tries its luck with a new supplier next season.