I cycle - I don't use the pavement unless it's a shared use one and often, don't even bother then since the road tends to be far quicker. I'll use 'shared use' one if it avoids a big one-way loop or a nasty narrow junction but not otherwise. Of course, not using the shared use pavement can lead to drivers shouting at me but can't keep everyone happy.
Licensing-wise - my only comment on it is that we've had it for years for cars/motorbikes and people still drive stupidly so will it really make a difference.
The reason most adults give for pavement cycling is lack of confidence or being scared of traffic and there is something that can be done about that. Councils run free cycle training that can be tailored to the individual. It's one-on-one rather than in big groups and the instructors will tackle whatever the cyclist wants - from a rank beginner who wobbles to someone who is confident about cycling but just doesn't feel comfortable on busy roads. Bike shops and workplaces should promote this more.
I will reiterate though that even inept idiot cyclists on pavements will do their utmost to avoid hitting anything (grannies/prams/litterbins etc) since they are likely to get just as (or more) damaged that whatever was hit. Falling off your bike at any speed can hurt. A drunk vagrant at Vauxhall fell into the cycle path as I was going along it... I'd spotted him being a bit wobbly so had slowed down but he then fell into my path. I swerved and came off mostly upright but still ended up with a badly swollen knee and a bruise the length of my shin that stayed for 6 weeks. Although the worst thing about the experience was him trying to hug me to apologise....