Stubborn black ink system

I'm trying to resurrect a machine that sat idle for number of years. Went through the Brother scripted routines including cleaning, purging and contacting Authorized service then tried the youtube soaks , cleaner injection etc but sill no black ink out of the head

I think a more complete understanding of the Brother ink management system would help me understand where the blockage is located and figure out how to clear it.

How does the ink get from the cartridge to the print head? Is it pushed from a pump at the front of the machine, sucked from the print head? or pulled from the front to an intermediate point and then pumped to the head? Or is it drawn by capillary action to the print head? Some other way?

What happens to the ink in the cleaning and purge cycles? Does the ink get pumped through the nozzles? Past the nozzles? How can the black ink cartridge be empty when there is no black ink coming out of the head?

When I pushed cleaning solution into the port that receives the cartridge, I expected the solution to show up on the paper towel pad I had placed under the print head. Instead it appeared to come out next to the rail on the side of the print head toward the front of the machine. Does this suggest a damaged print head?

What is the function of the four depressions in what appears to be a rubber block below the head in park position? Are these vacuum ports that move ink to the ink absorber pad on the other side of the machine when the machine is in a purge or cleaning cycle?

Is it normal to have a buildup of ink on the sponge at the left of the machine? When I started internal inspection I discovered an area of ink build up approximately 1/4" x 3/4" and more than 1/16' high. It seemed as though the printhead was building ink up in in that spot much like a 3D printer creates it's output. My conclusion has been that the entire sponge surface should be flat so that it can clean any wet ink off the surface of the print head. I successfully used cleaning solution to remove most of the buildup.

Apparently Brother does not make their service manuals to to DIYers. Is there someone who has taken these devices apart and figured all this stuff out willing to enlighten those of us who have not? Anybody know where there is documentation on the web willing to share a link?