Currently installed is a recent (about 10 years) Harbor Bay 5 blade ceiling fan with 44" blade diameter.

In a box I have a recent (about 10 years) Harbor Bay 5 blade ceiling fan with 52" blade diameter.

Yes, the large blades will run the motor harder.

Would these two fans likely use the same motor, so that I can just swap the larger blades and leave the current motor? Or is there a high chance the motors are sized specific to their blades, and I would need to fully swap the motor to use the larger blades?

The blades appear to use the same connection.