I haven't used Sainsbury's so can't really advise you but I thought it looked quite good.. It's all a very complicated tbh and designed to be that way I suspect...
I've had dogs 9 years now but only been insuring for the last couple of years. To be honest, I've been lucky and actually insuring before now would have been a waste of time and cost me more than any of the vets bills I'd had - that I could have claimed on. I'm paying £24 a month for 2 dogs now and I've just submitted a claim (my first one) for nearly £400 for treatment after an Adder bite BUT, I agreed to more blood tests etc than was probably necessary because I knew it was covered.
I'm still a bit in two minds about it all - it is expensive and we could afford to treat a serious problem if one arose, so I'm still not convinced it's totally worth it and we might just be better putting the money into a separate account every month. Having 3rd party insurance does give me peace of mind because that's a different kettle of fish...