Why can Neutral Red (indicator) not test values below its pKa?
The colour of neutral red in aqueous solution changes from violently red (protonated cation) to yellow/mostly colourless (the uncharged amine).

If the $pKa$ is 6.8, then the dye is still very much red at pH 6.8, and you won't be able to see the colour difference to, say, pH 5.

You have to keep adding base up to a pH of eight until the concentration of the colourful cation is so low you cannot see it any more.

Other indicators change from one strong colour to another, there the $pKa$ will be more in the middle of the transition range.