I heard Netherlands is somewhat strict on multiple / dual citizenship.
Plus if travelling to Netherlands as a citizen, they may require entering the country on their passport (similar to US and US citizens entering the US). It's definitely something OP needs to investigate, potentially with a lawyer in Netherlands.
Travelling to a third country - sure, I'd just travel on US passport for simplicity. Unless Dutch passport has some better benefits at a particular destination. For example, travelling to EU is going to require eTA soon. But OP won't need to do it as a EU citizen.