Unless the plug has been broken or pulled out I would just use a larger diameter screw. As I recall the plastic plugs you screws into are roughly about 1cm deep. The stock screws are a coarse thread screw (approx 4 or 5mm) like a wood screw without a sharp point. At 6mm it sounds like they holes are already oversized and stripped. I have used these products cut to length if the plug has fallen out.
https://www.zoro.com/zoro-select-thr...caAiUKEALw_wcB
The soldering iron is a good idea too but don't go too deep.
