most likely, there's no aerator or screen of any kind on the tub spout
That, and the tub draws a lot more hot water, which causes the dip tube to stir up the sediment more energetically vs smaller fixtures.
a hot water tank be drained/flushed-out annually to prevent the build-up of scale and sediment inside the tank.
Flushed, yes. Drained, no. When draining, the sediment just sits on the bottom because the outlet for the drain is slightly above the bottom of the tank. Flushing allows the rush of cold water coming in from the dip tube to stir up the sediment, so more of it can float around and have a better shot at being pushed out the drain.
I guess you could drain and
then flush, but it doesn't seem worth the extra effort IMHO.
Anyway, I like to flush in the spring or fall so I can run a hose out a window or door without the A/C or furnace working overtime. I run the hose to a concrete patio/walk/driveway and just let 'er rip until I stop seeing chunks shoot across the concrete.
BTW you should pull the little lever on the T&P valve at the same time to make sure it's not frozen.
I believe the faucet is a 2 control type (hot and cold have separate handles/valves). Given tenants description of the particles as "like black dust" I'm guessing it's most likely a gasket or ring in the hot water valve. I'm not thinking the lack of a strainer can explain this symptom
If it was a failing washer, wouldn't the faucet drip when shut off? A little washer can only make a few specks before it's all gone.
I doubt the tanks has ever been flushed in its 7 year life.
Ouch, most water heaters have only a 6 year warranty, and that's if it is flushed regularly. You might be looking at a new water heater soon...