Analog current and voltage adjustable power supply design
(1.) When the circuit is working properly, the output voltage is close to equal to the voltage on R1 center tap, because the VT2 diode voltage drop is roughly equal to Vbe on VD1. You have changed VD2 from a bipolar diode to a Schottky so you will have some temperature drift. (2.) You can certainly replace R1 with an adjustable regulator; you just need a circuit there to provide a variable base voltage. (3.) In your circuit, and R1 from 4.7K to 50K;  the base current for VT3 might be getting starved.  If you get rid of the pot and use your regulator as you proposed, it will probably help.  (4.) This circuit is designed for bipolars.  (5.) The 1k is necessary to provide current to keep VT2 biased on with no load; it is not necessary if you always have a load.