They are high limit switches, they open if the water reaches 135f, 160f. They are safety features so if the water after the heat exchanger gets too hot it shuts the system down. One of them presumably was bad that’s why it fired up when you jumped them. You have to figure out which one is bad by jumping them individually or replacing both. The pressure switch could be good if the piston hits the microswitch and the microswitch is working properly a simply continuity test would work. Jumping any safety switch should be done for testing purposes only and the jumper should be removed after.