John
Good description, just a couple additions.
Thatch is living roots and shoots not just the dead stuff. It can be thought of as a condo for disease and insects, the thicker the layer the bigger the population possible. The thatch will also slow or prevent water, nutrients and pesticides from getting to the soil. When thatch dries out it can become hydrophobic. If a thatch accumulation is buried by topdressing it can attribute to black layer. Some thatch is desired to help protect the crown of the plant, <1/4" and is usually referred to as a matt layer.
When to remove - when the turf is actively growing.
How often - depends on turf type, time of year and other cultural practices. If an aggressive topdressing program is in place it may never be required, generally verticutting and topdressing go hand in hand.
Verticutting assists in getting nutrients and water to the soil, and does allow the turf to breath as you put it, by allowing the gaseous exchange to occur of CO2 and O2 back and forth from the soil and atmosphere providing the soil is not compacted. V-cutting will also help firm up the surface by removing the puffy thatch.
To save $$$ by skipping a practice or 2 perhaps for a short time, but to eliminate the program will cost more $$$ in the long run. The super on site could be able to give you a hands on view and I would guess welcome your interest.