Hey there, I know I'm slightly late to the party (by a few years), but seeing this issue has not been resolved as of AutoCAD 2019, and we all would like to purge our drawings of inconveniences, I would like to offer a work-around for this issue that has helped me as of late.
Firstly, for this work-around to be effective, make sure you have well-defined text + mleader styles in your drawing. Make these the "Current" styles in your drawing (commands "STYLE" and "MLEADERSTYLE" -> browse for style and click "Set Current"). Important to have the box "Always left justify" turned off in your mleader style (see pic).

Next, you'll need the following LISP routine L2ML that allows you to select a leader object, then an mtext object, and merges both selections into a Mleader (and deletes the selections). The first selection does not have to be a leader object, it can also be a polyline. I found it over at https://forums.autodesk.com/t5/autocad-forum/mtext-to-multileaders/td-p/5336487. Included as attachment as well (you can always APPLOAD it into your client as well).
Now you're ready for the fun.
1) Identify infected Multileader whose text won't auto-justify when dragged with grips.
2) Explode target Mleader. You should get the arrow head, line, and mtext objects.
3) Engage L2ML command.
a) Select the previous leader line (which should be a polyline object).
b) Select mtext.
4) You now have recreated the mleader and once dragged, see its auto-justification property has returned.
Unfortunately, this work-around means you have to work mleader by mleader to restore its auto-justification. However, I find it is much more efficient than manually creating new mleader objects and pasting the text content into the new leader. You can just group-select all the infected mleaders, explode, then L2ML a few times, and finally adjust overall scale if necessary.