Why do we assume there's no cross immunity?
The scientific community don't assume anything- that's a defining characteristic of science. There is ongoing research into pre-existing T-cell immunity but no evidence of a big effect yet. The fact that the virus was able to spread so rapidly strongly suggests such an effect would be small.
they thought that may be ine reason why children are much less severely affected than adults.
It is possible but one of the reasons kids are snotty is that they have had fewer bugs than older people in their lives, so have developed less immunity.
Why would children be more likely to catch it for every year older that they are?
They may be a bit less likely to catch it but mainly it's that there less likely to experience severe symptoms. The prevailing hypothesis is that kids are less affected either because of Ace2 levels or because viral load/general immune system health makes a big difference and kids have plenty of phagocytes and a high metabolic rate.
There are only 4 other human coronaviruses out of hundreds of cold/flu viruses but the number of children getting severe cases is tiny. Also the effect of increasing risk is seen all the way up the age scale- if it were just about recent exposure then it would level off between 16 and 20ish. Also, kids get markedly different symptoms to adults, suggesting the factor at play is differing biolchemistry not pre-exissting immunity.
It seems very, very unlikely that this is the main way kids are so much less vulnerable.
Pigletjohn defeated in what sense?
I presume PJ is pointing out that sterilising cross-immunity is actually quite rare and only occurs between closely related diseases. The illnesses PJ lists were once common and serious but are now controlled through vaccination for that particular disease, rather than anything to do with pre-existing immunity.
It would be lovely if one of these speculative ideas were true and the pandemic was about to end because of T-cells or the disease mutating or injecting bleach or hydroxychloroquinine or faries or whatever. Given that hasn't happend so far the best plan is to listen to the scientific consensus and not get our hopes up about every speculative area of research. Humanity is explorin every posibility which might help develop treatments- most will be dead ends.