The "hood" above the center wire (electrode) and the "fork" at the top of the assembly are the flame-sense portion. Yours is extremely dirty. Use a "Scotch Brite" type scrubber to try to get it back to shiny. You can use a dollar bill on the electrode. Clean the mounting surface as well. Interesting that you were able to remove the pilot tube, they are usually very stuck.
Along with the high-voltage spark a very small electrical current is sent along that blue wire as well. When the pilot lights the flame (which can carry current) completes the circuit sending the current to "ground" and telling the system to open the main gas valve and stay on. That much dirt can cause enough resistance, especially as the metal heats up, that the control "thinks" the flame has gone out and shuts the system down to start a new cycle.