This question is specifically about presents left by Santa and not about his existence.
- Santa exists.
- He delivers presents to all good children.
- He does not have to deliver presents by traditional methods.
- Time is not an issue, Santa perfected time stretching decades ago.
- do a large portion of the children around the world get no Christmas presents?
- Parents not get confused by the extra present under the tree?
- Some kids do not get two presents from 'Santa' (one actually being from their disbelieving parents)?
Santa must deliver the presents, he cannot cede this duty to disbelieving parents. He is indiscriminate; it is not the child's fault if they do not believe in him, and the child still gets a present (presume childhood stops at 16).
Is there some evil force at work which steals the presents? If so, what does it do with them? Why hasn't Santa trained a force of fighter elves to stop it?
Children who do not get presents are not necessarily naughty. Santa's nice list will have everyone with a 'good heart' on it. Basically small bad acts (not handing in your homework on time every time, pulling your sibling's hair when they make faces at you, etc.) do not put you on the naughty list. So answers should not include most kids being on the naughty list.
I don't mind magic, but I would prefer a solution other than mind meddling/altering, this is a dangerous and presumably intensive process.
I would prefer a more physical reason, such as a villain opposed to Santa who steals presents. Maybe the villain gets stronger with an increasing amount of something? But why would the villain steal the toys?