I see you are in the Netherlands. I was recommended to use Ponal Wood Glue (used in Germany) for strong glue. It doesn't smell of anything and glues things really tight! Although it might be a bit of an overkill if it's just for popsicle sticks. I used it to glue legs onto a house, so it had to carry a bit of weight. As long as there is no residue showing it's fine when it's dry.
https://www.amazon.de/Ponal-PN18-Hol...keywords=Ponal
This one is the one that is generally recommended as a non toxic all purpose glue
https://www.amazon.de/Washable-Schoo...ords=Elmer%27s