I was thinking what Cody said, sounds like your blanking circuit is not functioning properly or is not connected properly to the driver board. When you power up the game, the blanking circuit should start LOW for about 1/2 a second which gives the CPU time to start up and initialize all the PIAs. The LOW blanking signal blocks solenoids and displays from operating during this period. Before the PIAs are initialized, the lines that control the solenoids and display segments are in an undermined state which could be on or off. As you are seeing, it appears that several of the solenoids are on, which blows the fuse.
Also, the blanking signal in most cases will go low if CPU stops functioning (defective ROM or socket, flaky CPU or socket, etc), preventing solenoids from locking on and/or burning segments of the displays while the CPU is in the locked up state.
Since your game appears to be firing the solenoids at startup, and since you can get the through solenoid test if you disconnect the solenoids for startup and reconnect after, we know blanking is HIGH on the driver board during the startup process when it should be LOW. The blanking signal is generated on the CPU board and as Cody mentions, it is transmitted to the driver board through pin 37 of the 40 pin inter-board connector. If the initial LOW blanking signal does not make it to the driver board through the 40 pin, the pull-up resistors on the driver board pull blanking HIGH at start up. In this case, the usual (incorrect) behavior is several solenoids fire at startup. On some titles, this causes the solenoid fuse to blow, others it does not.
What we don't know is if the blanking circuit is defective (stuck high all the time), OR if the initial LOW blanking signal is not making it to the driver board. The event we need to catch is during the first second, which is tricky as everything happens quickly at startup.
Do you have a logic probe?
Have you replaced the 40 pin inter-board connector?
If you have a logic probe, set it up, disconnect J11 (or pull the fuse that blows) and carefully monitor the signal at pin 37 as someone else turns the game on. You should see it start low, then go high. It happens fast so try it several times.
You can also try the same procedure with the logic probe on the blanking test pin on the CPU board. The blanking test point is just to the right of IC15 and J3 and J4.
If blanking does not start low, the most likely cause is IC23 (556).
Another less scientific (less exact) procedure that may tell us something is to observe the displays during startup. Do you notice garbage on the displays the first instance the game is powered up or do they start blank and in a fraction of a second start with fully formed numbers?