Why is it required to put an external inductor and diode at the output of a step-down switching voltage regulator?
Preface: we're talking about 'buck' (step-down) regulators here. Unless stated otherwise all comments apply to that topology only.
tl; dr: The LM2576 is a very old chip. Switching regulator technology has evolved over time, with further integration happening (including that pesky inductor). More about that below.
Why isn't the inductor part of the silicon?
In theory it's possible using MEMS techniques to make inductors (and some RF systems do exactly that), but power inductor values and currents are too large to be practical for silicon. So, the inductor lives outside.
Why the inductor at all?
In all switching regulators, the inductor is the energy storage element that converts a pulsed source into a smooth DC (along with the filter cap) to the load. In a buck regulator it’s basically working as a low pass filter (integrator), but its role is more subtle than that. In short, it's a way to transfer energy in one voltage domain to another, without the losses that happen with linear regulators.
The inductor is in one of two states:

charging: high-side FET is on, flux is increasing (storing energy)
discharging: high-side FET is off, flux is collapsing (releasing energy)

So the regulator IC's job is to steer that energy at the right time from supply to load, by steering the current path to/from the inductor during the regulator switching cycle.
For the buck regulator, the inductor input (left) side swings positive during charge, then negative during discharge. Why does it swing negative? Because of that collapsing flux resulting in cutting off the current from the high-side FET. That current has to go somewhere, otherwise that negative voltage gets very large.
Which brings us to the diode. Why?
The diode catches the inductor's collapsing-flux reverse voltage and forms a path for the inductor to dump its energy into the load. Unsurprisingly, it’s literally called a catch diode in some documents; and I'll keep calling it that here, because it's catchy...
And why Schottky?
Schottky diodes are preferred for catch diodes because they have a lower forward voltage (Vf) of 0.3V vs. 0.7V for normal diodes, and they have a much shorter recovery time (the time to switch from forward to reverse. More here: What is the reverse recovery time in a diode?) These traits improve efficiency.
Is there anything better?
You bet.
Some switching regulators include an internal low-side FET to catch the discharge, and so don't need a catch diode. These are called synchronous regulators, and they're more efficient since there is no diode forward voltage drop during the discharge-and-catch state. This type is popular and inexpensive now.
For high-power and high-voltage applications there are also switching regulator controllers that use external FETs, because the FETs are so huge that they too aren't practical to combine with the control circuit silicon due to the special process and packaging needed. Some controllers even support multiple phases for extremely high currents (hundreds of Amps for power-hungry CPU / GPU chips.)
Can I find a chip that has all this stuff in one package?
¡Sí, se puede!
For low-to-medium power applications, there are modules that integrate the inductor as part of the package. What makes this possible is ever-increasing switching frequency. The higher the frequency, the smaller the inductor value needs to be. So now it's possible to actually fabricate the inductor as part of the package leadframe (clever, huh?)
Several companies offer very compact solutions in the 500mA - 6A range that only need external capacitors for filtering. They're super easy to use, save board space, and in my experience, work really well. I've used them on M.2 cards for example, which have a strict height limit.
Here's a 6A, 17V-in buck DC-DC module that's only 4x6mm, that would replace the LM2576 in most applications: https://www.monolithicpower.com/en/products/power-modules/step-down/mpm3650.html