The EFF and similar organizations often voice their opposition against the patent system.

But why couldn't the EFF use the flaws of the patent system to their advantage.

They could file all kinds of patents for things they oppose. DRM schemes, surveillance and tracking tech, dark patterns, etc... And then patent troll companies that use these techniques.

Win/win. If they lose their lawsuit, it creates a precedent against patent trolls. If they win, that's one less of the things they are against.