TBH , you have every right to change provider, and if you feel that the way the BT ‘guide’ dealt with you was unacceptable, that’s reason enough to want to move ,I’ve no issue with your reason to leave , but the suggestion that as a leaving customer, you service is being sabotaged , is fanciful, imagine the outrage if that were the case , it’s not like there are no Ex BT customer service reps working for other ISP’s ,that could let the cat out of the bag , that sort of info would be priceless for competitors, and secondly, the timescale in switching, which again is not of BT’s making but your own decision to cancel first and explore your options later .
I hope you do have a good experience with Sky , that their service is working very quickly after the BT service stops , that Openreach fix whatever the issue is , and that your final bill with BT has a deduction for the time a fault was present on your service.
FWIW , my BT F2 broadband was £30 and is going up to £32.40 ish with the annual increase, today the equivalent broadband with Sky is £27 ( on an offer ) going to by £3.50 1st April , so £30.50 , so around £2 cheaper ….a saving , but not half price