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However, while cleaning I noticed a part that contacts the video head.
It appears to be a self cleaning part, but dirty and full of black
particles. At this point, it is getting the newly cleaned head dirty.

If you observe the tape loading in action, you will see that this
is bumped over against the head momentarily during the loading
process. It does tend to knock off some of the crud that accumulates
on the head cylinder, especially with rental tapes, but as you have
observed it can get dirty enough over time to become a problem in
its own right. If you can't find a replacement, leave it out. Newer
technologies are rapidly making tape obsolete anyway, and it doesn't
make sense to spend much money or time on VCR's.