Tesla coil - my primary circuit doesn't work
You have not mentioned what type of transformer it is. Solid-state transformers will not work for Tesla coils, as their outputs are a much higher frequency. They often look similar to the image below:



You really need the big, heavy, bulky type shown below.



If yours is a little, lightweight box then it will not work. From your description of the results I am willing to bet you have a solid-state one. With a higher frequency the capacitor will act more like a short circuit (as JRE mentioned in the comments) and it will not charge/discharge in the desired manner.

If you do have one of the big bulky ones, however, with a 50/60 Hz output, then my guess is that it has built-in fault protection. Transformers like this often shut off automatically if a fault occurs, which could happen if it detects an open output (example, if your spark gap is too far apart). It will need to be reset (unplugged and plugged back in) before it will run again. These transformers also do not work well for Tesla coils for this very reason.