Here's a link that explains what the EFE part of your system does.
http://dnr.louisiana.gov/sec/execdiv...p/auto/n/n.htm
I'm don't know a whole lot about it, but I THINK the EFE device that the vacuum switch controls is that butterfly valve in the exhaust pipe on the passenger side just after the exhaust manifold. When the engine is cold, the butterfly is closed to make the exhaust manifold get hotter faster, which helps with fuel evaporation, and also gets warm air going up that silver flexible hose on the passenger side to the air cleaner. It's all fine and dandy assuming everything else in the system is working correctly, like the little thermostat thing in the air cleaner housing that controls that vacuum flapper in the air cleaner intake snorkle.
All that said, I don't know if that EVE TVS that you broke sends vacuum to that butterfly valve in the exhaust when the motor is cold, or after it warms up. In ohter words, don't know if vacuum applied to the butterfly valve opens it or closes it, but I THINK vacuum closes it while the motor is cold, and that switch you broke cuts off the vacuum to it after the water gets hot.
In theory, everything in the system is suppose to be in place and working to pass inspection, well, at least here in Houston where they do the emissions test. But if you think about it, you truck is warmed up when you go to get it inspected, so the EFE system is already turned off when you get there. So bypassing this wouldn't make you fail the test, but they might no pass you if they notice it's been done. I THINK all it would do, if bypassed, is produce a little more emissions when the engine is cold, may cause the choke to open a little slower (if you're carbureted and have a choke).