Mummify and Shadow-court Satyr

So why does it not burn right away? It arrives on the minion without counters, and per its card text it burns when it has no counters.

Because the card is not played, it is simply placed on the Satyr so he can use it later. So all of the card's text is ignored/'inactive' until it is actually used.

There is a template for that is used on Inceptor, "face up but out of play". It is not used on the Satyr. Without it, what is the justification for considering a face-up card in the controlled region as not in play?

When you 'put a card on' the Satyr it is not played - eg: compare to Summon History ('put that card in play') or Concealed Weapon ('equips with a non-unique weapon card from your hand'). Until a card is played its text box is effectivelly blank.

If it were 'played' at that time then the first sentence would trigger before the burn clause, and the stick would get 5 counters and become a weapon immediatelly.

You are right that SCS does not use the same template as other cards with similar effects, but then again: when was VtES known for consistently sticking to templates?