No, the shaft of a bottom bracket that mounts the crank arm is correctly called a "spindle" as it turns with the crank. An "axle" is stationary and the rotating assembly rotates around it as in hub axles.

Now that the semantic are out of the way, I wonder if that bottom bracket is narrower than the bottom bracket shell. Say a 68 mm bb installed in a 70 mm shell. The excessive depth and inadequate lock ring engagement indicate that might be the problem.

OP: If you really need replacement cups or other parts for that crank and bottom bracket, you might post in the "C&V" forum here. Cottered cranks have been obsolete for decades and these people are well versed in their care and feeding.