Seems to me focusing too heavily on price in this matter could be a false economy.
You could wander into a couple of local estate agents and ask who they use.
That should get you a name or two of conveyancers who are at least competent.
True you're right, however, my father is quite frail just had 4 days in hospital, my sister who he lives with due to a burst blood vessel in her eye is unable to drive, they live 45 miles from both me & the property, they also live 15 miles from any solicitors offices. Yes I can drive over, pick them up, take them to a conveyancer & then come home all of which I am willing to do, but if there is a 'good' online one it would make life so much easier.
Buying/Selling a house although a big commitment should realistically not be anymore complicated than buying/selling any other item, yet solicitors/conveyancers seem to have been able to wrap it up in smoke & mirrors to make it as complicated as they possibly can.