I'm not sure what you are reading that is promoting fuses as running OL protection for motors, but it's just bad juju. In a motor circuit, you must have 4 main elements: a means of disconnect, a means of control, a means of protecting from short circuits, and a means of protecting from long term running overload of the motor. Those do not necessarily have to be separate devices, you can combine the elements, but they all need to be there and the devices you use must be qualified (listed) for the task(s). So for example a Circuit Breaker provides Short Circuit Protection Device (SCPD) and is ALSO a disconnect device, so it is 2 of the 4 elements right there. The same would be true for a fused disconnect switch; SCPD and Disconnect as one unit. If a Circuit breaker is ALSO listed as a Manual Motor Starter, it can ALSO be the controller (manual on-off control) and have an adjustable thermal OL protection too (I have done this), so you have all 4 elements in one device (so long as manual control is all you need). You can ALSO use a HP rated Fused Disconnect as the manual controller and SCPD, but if the fuses are sized to that narrow window to be within 125% of the motor nameplate FLC, it too CAN be all 4 elements in one.
As said, the SCPD can be fuses or a circuit breaker. Fuses in some cases are better than Circuit Breakers (i.e. when the Available Fault Current is extremely high), but always run the risk of single phasing a motor. If fuses are used only as the SCPD in a classic motor starter with a contactor and OL relay, it's a good idea to have Single Phasing protection with it to prevent that damage to the motor, and most of the new Solid State Overload Relays provide that now. But using fuses without something else to protect from single phasing, as would be the case when trying to use them AS the OL protection too, exposes you to that risk.
But just because you CAN use fuses in lieu of an OL relay, doesn't mean you should and there is no cogent argument to doing so, other than a "penny wise - pound foolish" approach to trying to save money on component costs. You can shave a few dollars from the cost of a motor controller, but you risk a much more EXTREME cost of lengthy down time and rewinding an expensive motor. Not worth doing, no matter what the fuse manufacturer wants you to think.