Table saws are typically equipped with AC induction motors, also know as 'asynchronous', or 'squirrel cage' motors, if you've heard any of those terms.
First things first: induction motors are magical devices. One of their many magical properties is they draw a variable amperage based on the load you place on them. They also have upper and lower bounds, call 'unloaded' amp draw and 'full load' amp draw. The plate rating is going to be your FLA (full load amp draw), or what the motor would draw if it was loaded to its maximum rated horsepower. When you started your motor, it drew 4 amps. That's unloaded. At that point in time, the only power the motor is using is the minimum amount needed to keep its stator field from collapsing. Typically, unless it's a huge industrial motor, this is just a few amps. Your unloaded amperage is higher than I'd expect for a single phase 3/4 or 1hp tablesaw motor, which is probably an indication of motor windings in subpar shape. It is possible to have motors rewound, although that's not really cost effective for a motor that small.
Second thing: ac motors don't have brushes (which is why they're so quiet), so there aren't any easy parts to swap out.
I'd put a new blade on it and see how it does, and it if still struggles to cut through greater than 1/2" material look into replacing the motor. Depending on the age, there may be a dedicated online community that are experts and refurbishing/repairing that exact saw.
Good luck!