I don't follow that, in the event of losing balance away from yout standard right leg there is typically going to be verge or pavement normally rather than passing traffic. Positively there may be a fence or wall which will stop the overbalance.
In UK traffic if you're left foot down, right on the pedal, youre leaning away from the traffic flow. If you fall you will go in the direction you are leaning - physics makes it very difficult to do otherwise - and you will land on the verge/footway/sleeping hobo (delete as applicable).
If youre right foot down in the UK youre leaning
towards the carriageway. If you fall you will will fall into the road, which isn't a good thing.
Most people lead from the right leg and I've often wondered how that factored in countries where folk drive on the left as it would mean starting from the weaker leg for the majority of riders. I only ever taught advanced road skills in the UK so never got the answer to that one. I am still an IPMBA member so could probably find out if anyone is really interested.
Never thought about it until now.
I walk my bike on my left hand side. Mount and dismount from my left hand side.
There's two possible reasons for that.
I'm left handed and it feels natural .
But I really got into cycling when I lived in Europe. There it's the safe side to avoid dismounting into traffic.
Thinking back, 60+ years, when cycling to school you had to dismount from the right.
That was so you could raise your cap to the master on gate duty and guide your bike.
My things have changed
Where possible when pushing keep the bike between you and the traffic. Again, if you fall it makes it less likely youll end up having your head run over.
If youre not alongside the road then push from the left, the UK nearside. If you fall or drop the bike theres less chance of becoming entangled in the drive train, which had been known to cause very serious injury for the unlucky.
As I mentioned above, some of my students were wired up in such a way that they struggled with this sort of thing. I didn't used to get hung up on it. It's best practice and not the ten commandments. Just so long as riders are aware of the consequences, and thus thinking carefully as they go, its not such a big deal.