All public sector workers get paid for days off sick. IIRC (well it was 20 years ago) they are entitled to six months continuous sick leave at full pay, followed by six months half pay.
The word "all" is dangerous. Locum doctors and supply teachers don't get paid if they don't work, but almost all workers get paid for days off sick. My public sector contract is exactly what you state: six months at full pay and six months at half pay.

I am not sure where this idea that throwing a sickie is worse if you work in the public sector. Whatever way you look at it, you are defrauding your employer and this is equally bad whether you work for the government, a multinational or a tiny business.

If you read the union website carefully they don't advise their members to pretend to be ill; they just tell them what might happen if they do, i.e that they will probably get away with it, but might be in serious trouble if caught. Of course they should advise their members not to throw a sickie because it is theft and also because other union members, maybe the ones who don't like football have to cover for them so it is unfair on colleagues too. This piece of advice is missing, unfortunately.