So the only relevant congestion for this specific issue is between the tunnels and the highways. Manhattan residents living near the tunnels know what they're signing up for. The tunnels opened in 1927 and 1937. The traffic isn't exactly a new problem.
Also, the parts of NJ in question (where the tunnels are closer than bridges) are largely urban in character. You make it sound like people are coming from some far-away leafy suburb, that's not the case for the majority of this population.
Any congestion pricing plan which ignores that problem is going to be met with mass outrage, to an extent that will swing elections against incumbents. Especially when said incumbents promote this "no charge if you stay on highways" BS without explaining the fine print.
Again, I'm saying this as someone who only ever takes public transit and is generally in favor of congestion pricing as a concept.