What would you have done if instead of a codgerbuggy it was a broken down car?
Or a horse/Deer/Bike that had crashed, large pothole, tree (or large branch)...
I vaguely remember my instructor (many moons ago) mentioning driving at a speed in which I could stop if something unexpected was in the road ahead, and that the limit didn't mean I had to drive at it if the conditions didn't allow for me to do it safely (including being unable to see far enough ahead to stop if there was something in the road).
I'm all for the scooters requiring mandatory tuition and insurance, but they are legal for road use if they have lights and reflectors, same as bikes - although minimum speed restrictions do apply to them (if a road requires vehicles to be able to reach a minimum speed for example).
Knowing how much some users do rely on them, I certainly wouldn't want to ban them, just try and get the users to know a bit more about them before using them (I don't think they are any worse than cyclists, motor bikes or car drivers in terms of stupidity levels, and at least their top speed is such that they are unlikely to cause serious injury to pedestrians).
I wouldn't even want a ban of their use in shops, as long as the user is sensible most large stores are big enough to safely manoeuvre a Class 2 with ease, and even many class 3's (they aren't much/any wider than things like wheelchairs or shopping trolleys and can turn in the same sort of space).
Re overtaking them... They are at most about 6inches wider than the average bike + rider, as they are governed by exactly the same thing that governs the width of most bikes - how wide the riders bum is

At most a mobility scooter will tend to be 60-75cm (2-2.5 foot) wide all in*, as they are designed to fit through the average house (or shed) door, so if used sensibly you shouldn't need any more room to overtake than a push pike.
There are a lot of them round here, but only one causes any problems on the road (an idiot who has been knocked off at least once), and maybe a couple on the pavements - although it's more common (from what I've seen) for people to step in front of them without looking (or step back into the path of them), than for the rider to be at fault.
The problem in some areas is simply that whilst their may be roads (NSL) connecting say a small village or houses with shops, there may not be any pavement, and whilst humans can usually manage the verge on foot, the scooters can't (let alone if the person was in a wheel chair), which leaves them in the situation that they can either use the scooter on the road, or potentially get stuck at home all day.
When I'm driving I have much more trouble with pedestrians and cyclists than scooter users

Cyclists because they have a tendency to pop on and off the pavement with little or no notice (something scooters simply cannot do), and pedestrians because they do much the same thing, and those with buggies seem to forget when they wait to cross that they've got a buggy stuck out in front of them
Having said all that, I don't agree much with the policies of some shops that have "loan" scooters, as IIRC they are often both larger than standard**, and the stores tend not (AFAIK) to have any way of telling if the user has ever ridden one before (it can take a few minutes to get used to the different handling on various models even if you've used one before), which makes me much more wary of them than I am of normal scooter users.
Proper "shop mobility" schemes do I believe require the user to register, and have them demonstrated (and tend to use more normal scooters often supplied locally).
As I say, better training and insurance for them would probably go a long way to solving many of the issues with the scooters, and some more understanding and patience from drivers would possibly solve other problems.
*Any less than about 50cm wide and you won't fit an average sized person on them without overhanging arms and shoulders (iirc airline economy seats are about 18" wide), and has serious stability issues. Out of the various models i've seen and my mum has used most have been max 60cm wide (including the class 3 that she found she didn't like).
**The ones at my local Morrisons and B&Q are large (compared to normal ones), 3 wheel based designs, high manoeuvrability but relatively unstable, and much heavier build.