Apart from Virgin all of the copper/fibre network is the responsibility of Open Reach regardless of which ISP you choose.
I don't necessarily agree with your concerns about contracts as they can generate good discounts and most are only for a year. Are you anticipating moving? If you suddenly decided to move it would most likely take 5 - 6 months at least and most ISPs would treat you sympathetically.
When choosing your ISP not only should you be considering customer service but also whether or not the ISPs uses traffic management, has unlimited access, has high set up costs and how good is their router.
I have found that the best way forward, if you are satisfied with your existing supplier, is to check out their competitors prices in your area then speak to your provider. I have done this with Sky for many years and they have also been happy to negotiate.
Finding the best customer service is always extremely difficult, for example, I am with Sky and various surveys rate them as terrible, which is confirmed by their forums. I have been with them many years and have always found that the little contact I have had with them has always been excellent. You may well find the same with SSE.
I have checked out SSE on line and they certainly aren't the cheapest or rated the highest for anything.