Bumping in hopes you find some help.

I would ask the county board of health to test the faucet.

There must be agency in your local town/city government that provides oversight with this kind of thing. If this is a rental, this might be in the Dept. of Housing. The town where I live will send someone from the Department of Health to collect water samples to be tested by a lab if this is requested by any resident. In your case, you might have them test the faucet itself as well as the water. Do not give up! Wishing you all the best with this.

Google manganese bacteria and see if that resembles what you are seeing. I I’ve got a similar situation with bathroom faucets. I first noticed it when I replaced a 1960s vintage faucet with a new one in 2000. The other 60s faucet in a bathroom and the new kitchen faucet never developed the black gunk. My internet searching back then led me to suspect the faucet body was made with lower copper content brass than old faucets or more plastic parts. Copper has some antibacterial effect. Our current house was built 15 years ago and all the bathroom sink faucets develop at least some black gunk. The kitchen faucet does not not develop it here either. Hopefully someone can shed more light on this.

I also read about doing something like that but with vinegar.

Jalarse, try wrapping a rubber band around the strainer, then use pliers to try and twist it. Hope this idea works for you. Where we used to live we had really hard water and I would have to change or clean that screen frequently.

Anytime you see black/brown gunk at a faucet or an undermount sink, it's leaking. Guaranteed.