Circuitry of TV splitters

You've made some wrong assumptions about what the parts of the system are. The part you're describing as a non-powered passive splitter is actually a powered active splitter. Splitting one aerial signal into six with a passive splitter is unlikely to give you sufficient signal on any of the six outputs, especially if the aerial is in the loft.

The part you're describing as the booster is just the power supply to the active splitter. It sends a DC voltage up the cable running to the active splitter, and filters it out of the cable running to the TV. That's why it has to be powered up for any of the TVs to work.

The rooms where TVs don't work are either down to a faulty output from the splitter or a defective cable. To do a basic test on each cable, disconnect it from the splitter and check that it's open circuit, then short one end together and check that it now shows a short circuit at the other end.