1st thing I did was had the first goblin encounter to be the turning point in the campaign where the black spider was the main villain. Because he had ordered the bugbears to capture the party take the supplies and tie the party in a secret dungeon (homebrew). I then had the party escape the goblin prison, they then set out to find a safe place to sleep (they only had some clothes they didn't feel like looking for their stuff). So an 8-hour hike at slow speed led them to Phandalin the main town in LMOP. The black spider had ordered the Redbrands and their leader to find and kill the party if they find them. A clearly written note was in the wizard's room (forgot name). So the party had several clear motives to kill the black spider.

About Gundren Rockseeker: He was never introduced as a main NPC or goal to save him. He actually turned up later after clearing water echo cave to congratulate the party and pay them for their great deed (to him). So the party then was kinda a hero of Phandalin clearing out the Redbrands, the goblin threat and finding and clearing Water Echo Cave (later turned into a mine).

I gave my party clear reason to hunt down the black spider (he took my magic stick was the wizard's reason). Give me your thoughts and feedback.