There’s a cycle lane and yet you think it’s ok to cycle on a path because you’re unsteady?! So, a designated safe spot for you to cycle but you’d rather put pedestrians at risk because it makes you feel better?! 
Earlier today my 3 year old was nearly knocked over by not one but THREE cyclists going at speed on the path. I have mobility difficulties and couldn’t get there fast enough to grab her out of the way (she was walking about 5m ahead to go hold her dad’s hand instead of mine and walking with her older brother). The first idiot startled her which made her side step out of his way and then the other two lunatics whizzed past one either side! She was hysterical at what happened. But yes, don’t you be worrying about children, elderly, deaf, blind, disabled etc as long as you feel better not cycling in a marked area!
Your 3 year old child cycling on a pavement is, IMO, fine as long as she is taught to go slow and stops to allow pedestrians past. You however should be either cycling on the road/cycle lane beside the pavement not ON the pavement, or walking on the pavement and pushing your bike.
The only exception would be on a totally deserted footpath that goes along a busy/fast national speed limit type of country road where there are never/rarely any pedestrians, there are no designated cycle lanes and where there would be a serious danger to using the road as per the Highway Code.
If you’re still confident that you are right and nearly everybody else is wrong then just ask yourself whether you’d be ok with a cyclist whizzing past on the bike with your young child walking close by and if you’d be ok with them knocking her over and potentially seriously injuring her, or worse, because she’d stepped to the side to avoid something on the path or had seen something exciting so she’d pulled towards it etc....if the answer is no then DON’T DO IT!
I can’t believe that you even have to ask whether it’s ok to put others at risk by cycling on a footpath when there is a designated cycle lane there to use 