Is it normal to be limited on exercises by assistance muscles? Should I do anything about it?

On bench press I always stop due to my triceps being too tired to do anymore. Yet my chest has never been sore or been what stopped me doing more, isn't that a bit crazy? I can feel it getting a pump after the workout, and it is growing but I've never felt anything

On chin ups it's the same story, except my biceps/forearms are always tired and I never feel my back is sore enough to be the sticking point. I get this as the lats are so much bigger than the biceps, but the chest is a bigger muscle than the triceps...

Should I look at adding some bench press where I only do the bottom chest-dependent part of the movement? Or just deal with it as it's normal?