Having the thermostat on the wire is fine, but you need to move the back of the thermostat as close to the bulb as possible. I'm talking about 1/16 or 1/8 of an inch. I built 2 incubators from a small refridgerators, and it took me awhile to figure that out. My thermostat is on a screen like yours so I could mount it out next to the bulb. That is what you want. But you got to have it close, even touching probably won't hurt.
When you have the thermostat too far from the bulb you can get one of two results: Either the air will cool down faster than the thermostat and will not turn on causing lower temps, or the thermostat will cool down faster than the air and come on too much. A lot depends on fan placement also in that circumstance. By placing the BACK of the thermostat up against the light the temperature from both will be the same. Also, I have a computer fan pulling fresh air in from two small holes, plus a small 5 inch fan sitting inside pointing up at the bulb/thermostat. This is just to make sure the air is moved around keeping even temperatures.
This is the most common problem most people have with getting their homemade incubators to work properly. Some people have a problem understanding that close means CLOSE! They put it about an inch away, and think, boy, that is close. But you might as well have it screwed to the outside of the incubator!
Mine isn't touching, but I don't know if I could get a hair between them.
Sometimes, depending on certain factors other set ups will work, but it isn't often. Try what I said and see if you don't get a much better temperature range than what you are getting. Mine is staying within just a few tenths of a degree.