Since it was me who mentioned that I belive I should clarify things - the head cover is not strictly required for the drive to operate. It only serves to protect the upper head from external EM noise. In some cases it also doubles as a dust cover, in which case it's a bit more important to have it there. However even in that case it should not causing damage to the floppies once the dust buildup is cleaned (a simple blow of air might be enough for that) and the computer is pretty clean inside.
If your floppies get damaged, and it's always the upper side, it might be there is some dirt already stuck to the head surface. It can be hard to clean without removing the head but it can be done, the most important thing is to be able to see it - to tell how bad it is and how the cleaning is affecting it. Another possible cause is the head being chipped somehow (though the plastic is pretty hard to damage) or the suspension spring is so bent that the head tends to rest on the media with one corner of it pressing pretty hard. The corners are rounded but still, it will cause damage like that after a while. Again it's difficult to be sure without disassembly, you need to be able to look at it with some magnification and see it if clamps down evenly vs the lower head.
If you get media damage on both sides then perhaps your floppies are to blame. It can still be dirt on either head but some floppies, especially if stored in a place with high temperature swings, can start shedding the magnetic coating. Which will foul the heads and only add to the damage over time, even on good floppies.