Thus far reading section 11.2, I could not understand what is wrong and why the RA3/RA2 arrangement via pot does not work.

To keep options open I was playing with LM324 arrangement. One issue I cannot figure out. With LM324 inline with the VU meter, without any signal being supplied to the LM324, I have 16 blocks on the top and bottom lines of the display. Once any signal is applied to the LM324, the blocks disappear and the VU meter operates as designed.

The sensitivity of the VU meter with LM324 is 300 mV full deflection of the VU meter scale on frequency of 1 KHz, or about 30mV per each block of the display.