Can see it would not be nice to work somewhere like that.
I haven't seen anyone on this thread who is saying they report children or adults seen out whilst off ill. They might judge them, but no mention of reporting.
Tbh Giles you sound a bit paranoid that people are out to get you - either at work for being off ill and putting pressure on you to be in, or to tell some kind of authority about you or your children.
I accept that it could be possible to be really ill in the morning and by the afternoon or evening to be feeling much better. However, I think it is unusual for someone to be ill enough not to be able to carry out their job and then to be so significantly better so quickly, so they can be out shopping/partying or whatever....so not impossible, but certainly not the norm .......which is why people might be surprised by it.
But the key issue isn't what other people think anyway. The key issue is for people to be at school or work when they are capable of working and not there when they are not. In most cases, if someone cannot do the work required of them due to illness, they are unlikely to be fit for shopping or partying later.
I think that if a family are regularly taking time off work/school and then involved in social activities later in the day, it would not be unreasonable for an employer or school to question them about the illness.
It would be unreasonable to put pressure on someone taking their first day off sick to come in, to allow an atmosphere at work where people discuss whether people are skiving. It is important that people are allowed the time to recover when they are Ill and not feel criticised for it. It is also important that people don't take off the sickness when they are not ill or the illness is mild and means they could easily work. It is people doing this which often makes it difficult for people who are genuinely ill to take time off, and why firms often have policies such as no pay for the first 2 days of illness/HR meetings after 3 periods of absence in a year.
Most people taking time off are genuine. But people do take the piss too.