24Vdc to 5Vdc 3A Buck Regulator over-heating
Problem: The LM3489 chip (supposedly rated for max current of 4A)...

The LM3489 is a PFET Buck Controller. Maximum current is determined by which external FET, inductor and flyback diode are used.

the only example circuits that I found demonstrating the LM3489 show
between 100-500mA outputs, nowhere near the 3A that I need.

You can use the LM3489 in a higher current design, but you will have to upgrade the external components to handle it.
The FDC5614P supposedly has a maximum current rating of 3 A when when mounted on a 1in2 pad of 2oz copper on FR-4 board. In practice you would not want to do that because the FET would run extremely hot and have a high chance of burning out, as well as heating up other components around it. In your circuit the FET should only be conducting for a part of each cycle so you might get way with it, but for safety I would choose a device with much lower RDSON.
If a 22 μH inductor is appropriate for 0.5 A then for 3 A you may need a smaller value to avoid saturation (4.7 μH is a typical value used in regulators with similar sized inductors).
When converting from 24 V to 5 V the flyback diode will have to pass ~80% of the average output current, ie. 2.4 A at 3 A. The MBRS140 is only rated for 1 A.
To determine the best components for your application I suggest you either simulate the circuit, or copy an existing design with similar specs. Ti's reference design PMP7771 uses a FET with 0.021 Ω RDSON, a 5.6 μH inductor, and a 3 A Schottky diode.