There's absolutely no reason to change the controls unless they are damaged beyond repair. It's an old machine, but that doesn't mean that old parts don't work and work well. You wouldn't graft the spindle from an Chinese mill drill on the machine just because spindle is newer, would you? The electrical engineers who put the motors and controls on these machines were every bit as good as the mechanical engineers that designed the rest. In fact, most of those old EEs knew a heck of a lot more about motors and such than most control engineers in the field today. The fact that so many of these old machines are still running just fine on their vintage motors and controls is a testimony to how well the old EEs did their jobs.
The difference between the overload devices and the type of circuit breaker that you find in your electrical panel is the the overloads don't operate to directly interrupt the circuit. Each one has a switch that is normally closed and opens when the heater reaches a certain temperature. The switches for all of the overload devices are wired in series so that if any one of them reaches the overload state the main AC contactor will open and shut down power to the machine. Emergency stop buttons are also connected to the circuit. For example, it the feed motor overloads, you don't want the spindle to keep turning and vise versa. Imagine if the spindle stopped turning and the feed motor kept running? Somethings going to get broken.
On large single phase motors the motor starter is a relay or mechanical device that connects a start winding and usually a starting capacitor to power long enough to get the motor turning and then automatically disconnects the starting circuit. On most machine tools a different type of "motor starter" is used. They are basically big relays (electrically operated switches) and are often called contactors. Pressing the start button momentarily applies power to the contactor's coil circuit, causing the contacts to close if nothing is wrong. Now, all of the overload's switches, the emergency stop button(s) and any interlock switches have to be closed in order for the coil to get power and cause the contacts to close. An example of an interlock switch is a spindle lock interlock that prevents you from starting the machine if the spindle is locked. When the contactor closes one of its contacts is usually used to continue to power the coil circuit after the start button is released. This same system is used on all sorts of industrial machinery, some so large that you can't see the other side. If someone has pushed an emergency stop you have to go and pull it out before the machine can be started.
The 3A fuse only powers the work light and the contactor coil circuit, not the whole machine. The machine's motors are protected by the overload devices. Often there are also fuses and a knife-type disconnect switch in a box on the back or side of the machine where power comes in, but those fuses don't provide the primary protection for the machine's motors.
If any of that doesn't make sense, holler and I'll try again.
Cal