Why is my LED flickering when it is connected to a variable resistor, and it's resistance is at the minimum value
There can be three reasons for this.

1, the slide pot is damaged or dirty at the top end. This could be dirt or something that is causing a physically bad connection, or the wiper of the pot is carrying too much current and causes it to flex. There is a maximum current it can carry before damage happens.

2, the led is drawing so much current it overheats and there is internal damage to the led die or bond wires and unstable flashing is a sign of this. At one end of the pot, it will be whatever resistance it should be, either 10k or whatever. The other end will be almost zero. So your overpowering that led.

3, the regulator is going into protection mode, along with reason 2, the led drawing too much current. So the regulator shuts itself off, then the led stops drawing power, then  the regulator turns on, rinse lather repeat. This explains the lower voltage your meter is reading because it's happening too fast for the meter to get accurately, and the flashing.