Slow bathtub drain. Tried plunging and snaking, no luck.

I have a slow bathtub drain. I blocked the overflow and tried plunging. I also have a metal hand snake that I tried to work through from the overflow but I think it it’s getting caught in the trap. I included a picture of the pipes from underneath. When I plunge, it looks like water is coming out of the vent, as I can see where water has run down.

What’s my best course of action? Could I try covering off that vent and plunging again?

Do I need to remove the trap and drain pipes? I’d like to avoid that if I could.