So why they wrote use 5w50 and not 15w50
You ask
two questions:
1) "Will 15w50 "damage" the engine. Answer is: NO, Kymco says it in an OK "alternative".
2) "Why Kymco say use 5w50?" Probably best to ask your local Kymco dealer.
But, again - previous owner did no damage with his 15w50 oil if changed at proper intervals.
Stig