I have been told I could improve my runner bean crop by planting seeds
in
cardboard tubes, such as toilet roll cores, to give a better root. Has
anyone tried this and does it work ?

Yes, that's what we do. Also the cores from kitchen towel. (We never have
enough, but you can get rolled black paper tubes from the big Sheds to make
up the deficit).
If you stack the tubes upright in seed trays, loosely tied around with
string, they are easy to water and support each other.
We start all our runner beans this way and also half of our sweet peas. (We
put sweet pea plants in tubes in the ground at 10inch intervals with a seed
beside each. This gives a dense and long lasting display .... it is still
looking magnificent right now despite the poor summer.)