The reason that doctors will issue sicknotes for illnesses under 7 days in length is that some employers will (illegally) ask for them before paying sick pay.
Yes, and the reason that doctors are not contracted to issue sick notes prior to 7 days is to try and prevent the surgeries being flooded with people with sore throats demanding a sick note for one day or two off work, because otherwise they'll get it in the ear from their employer. While there are a lot of bad employees out there, they are matched only, I'm afraid, by the amount of terrible, awful, tyrannical employers.
QDM