Electronic – Why is the Schmitt Trigger output fed back to the current source in NE 566 (VCO)

What you describe is workable - it's essentially a buck converter - but your exact circuit is not specified and there are many ways to go from a somewhat correct general concept to an incorrect circuit.

The design note 188 that you reference is about 15 years old and its time has passed. While it is a marvel of improvisation, you can get better results for less cost using modern ICs.
I'm aware that you only cited it as an example, and that you wanted a switch mode version, but using a dedicated switch mode converter IC would be a good idea at minimum, and using a LiIon charger IC would be an even better one.

An LTC1541 - datasheet here costs about $3 in 1's. It's quite a useful IC if you need its features. With a 5uA typical supply current it could not fight its way out of a wet paper bag, and a slew rate of 8 mV/uS means that you have to be clever to use it and the characteristics of the device can greatly affect circuit performance.

AN188 loosely uses the term "float voltage". A Lithium Ion battery must not be "floated" in the sense the term is usually used. They intend it to mean "hold at CV while charge current ramps down due to battery chemistry action" - but the applied voltage should be removed once current has fallen to some preselected percentage of Imax.

Once you have terminated charging there will usually not be any need to restart charging until there has been some load current drawn. This is because an unloaded LiIon cell will maintain a close to full state of charge over considerable periods. If it is going to stand unloaded for weeks or months then testing battery voltage for possible restart of charging may be in order - but the load from the voltage measuring circuit can easily exceed the self discharge rate.

If you are trying to produce a low cost design with a high element of DIY in it you could try the venerable MC340-63 switching regulator IC. These are about as cheap as you can buy a SMPS IC for and can implement almost any topology. As you have noted, to make a constant current driver you compare the drop across the current sense resistor with a reference voltage. The MC34063 has an unfortunately high reference voltage (about 1.2V) so you can use an opamp or comparator. This adds cost and complexity, but using an eg LM358 dual opamp is almost as cheap as the MC34063. Scale current sense resistor with opamp up to reference voltage. Figure 6. in this applicationnote give the general idea - in this case opamp inverts and it's a CUK converter, but the principle is the same.

The schmitt trigger inverter is what generates the signal that charges and discharges the capacitor. Assume the input on the left starts at 0V; the schmitt inverter will therefore output +5V. This +5V will charge Ct via Rt until the voltage crosses the schmitt trigger's rising voltage threshold. At that point, the output will change to low, and start discharging the same capacitor until the schmitt trigger's falling voltage threshold is crossed, starting the cycle over again.

Power for the schmitt trigger's output voltage comes from its power supply - marked +5V in the diagram.