> Isn't that because those issues usually aren't visible before around that time? (I am neither a GP nor a psycologist though so this is just trying to apply common sense.)

In some cases maybe, but it don't believe in all cases. A neighbour had this with her 7 month old baby, and symptoms showed up literary 4-5 hours after vaccination (not autistic symptoms of course, problems with breathing and similar), but at about 3-4 years you could already see some of the autistic symptoms in the kid along with speech impairment.