A tub flows at a much higher gpm than any other fixture in the house and an upflow filter has a much lower gpm SFR than the same cuft size downflow filter.
So when you ran the tub, you flushed some of this dirt out of the filter. Had you used the tub less frequently, as time went by, you would have seen the dirty water at other fixtures. It had nowhere else to go as it built up.
The filling the unit totally full reduced your flow rate through the filter because there was no space for the mineral to expand as you used water.
A backwashed filter would use like 50-75 gallons to backwash once every 4-6 days. That's hardly enough water use to cause a well a problem. If you added another person to any of the houses, that one person would use 60 gals PER DAY; would anyone say that no one could have any more family members because the well might not keep up? I think not. Plus, the 50-75 gals will be used in two batches of half the total gallons; the first for like 10 minutes at say 4-8 gpm, then a short pause and then the rest over the next 4-8 minutes. The gpm is less than your tub uses when you fill it. About the same as a clothes washer when filling.
The brown is iron that as it sits in the mineral for x minutes/hours and the pH is increased, the ferrous iron is oxidized into particulate matter (ferric iron-rust), causing the brown water in the mineral. Then you run water, the brown water comes out the filter and through the line into the tub (with your highest flow rate gpm). Then the water goes clear because the iron can't oxidize in the filter because there hasn't been enough time for the oxidation process to take place. When there is sufficient time, you have brown water in the mineral, and if you run a high enough flow (gpm) it is flushed up through the mineral; the mineral expands with increased upflow rate; meaning less filtration as the flow increases, and you get a slug of dirty water.
If you want to prove this, don't use the tub for a few days/week. Then one morning first thing, run water at the sinks and then flush the toilets all at the same time, and look for brown water in the toilet bowls and tanks and sinks. Let us know the results.