Safety, rationality and to encourage more forms of green transport. Most roads without designated cycle lanes are simply too dangerous for cycling.
For safety reasons cycling should generally be permitted on pavements unless designated otherwise in high density pedestrian areas. It is irrational that invalid carriages/scooters should alone have this dispensation since cyclists are equally vulnerable road users.This would encourage parents to allow their children to cycle more readily and might encourage cycling more generally.
In many out of town centre areas where pavements are often infrequently used by pedestrians I would suggest that the general rule should be dual use but that cyclists 'Give way to pedestrians'.