Sometimes you just need to peal back that black protective square (tape) to inspect for oxidation under the Thermo & ECO spring loaded safety switches... then clean the contact points with sandpaper (metal sand paper is preferred over wood sand paper).
Note: You will also see these spring loaded contacts have an insulation plastic spacer that you need to rotate (precariously under the metal tabs) into proper position it or your switch will not work.
These spring loaded switches are normally closed thermal safety switches that make a serial connection. This means you an use a jumper to bypass the switch in order to diagnose if the switch is bad or not.
All that said, my guess is that when you messed with the "brown wire" you moved the switch just enough so the contact with the tank was improved.
Therefore, you still have work to do -- clean the bottom side of the switch where it meets the tank or you will be without hot water again soon.
This diagram is specific to my Atwood A10-4E (10 Gal) Water Heater, but the function of these to spring loaded safety switches is common to all.
Also, that brown wire thermal-diode fails more often than you think and many people carry a spare, and a spare set of tank sensors, because you know these things will fail while you are camping! (So who you gonna call then?) I.e., having a spare set is not just about making sure your DW gets a hot shower... it's also to make sure you can get a hot shower too!
Note: If you use a jumper to heat your water be careful not to overheat your tank. Just 5 minutes should heat the water enough to take a warm shower. Then turn the power off and remove the jumper. Never leave the jumper connected... unmonitored.
One more thing: Some people think their hot water circuit is faulty if they do not have hot water. And they get really confused when they know there is hot water in the tank as evidenced buy pulling the tank water pressure switch.
Consequently, if you know you hot water in the tank an it has reached a temperature of 140F, then your problem is most likely a stuck check valve screwed into the back of your tank. ...And you may have one or two check valves, but it's usually the hot water check valve ON THE TOP that is the problem.
To fix a stuck Camco check valve - just remove it (if you have a big enough wrench) and then drill out the check valve and turn it into a pipe nipple. Then reassemble.
You do NOT need a check valve for your hot water to work. These Camco check valves are junk and cause a number of owners to end up without hot water for days.
Plus if you drill out the guts of that Camco check valve you will save yourself a $200-$250 repair bill.
A much better place to install a check valve is just before your hot water "T" to the shower line... where the hot water tank water line goes to your bathroom sink.
Read this for more information on where to install a shark bite check valve and why:
https://www.winnieowners.com/forums/...ue-359595.html