"Disregarding heavy muck, how many washes are needed to get an acceptable condition?"
From an instiutional/commercial laundry perspective things are usually broken down by soil level, light, medium and heavy..
Light soil - flush, wash, rinse or rinses and extract.
Medium soil - flush, wash, drain, wash, drain, rinses then extract
Heavy soil - flush, wash, drain, wash, drain, wash, drain, rinses then extract
For moderate or heavy soiling (depending upon type) some places will do two or three flushes before going to first of three washes.
Laundry heavily fouled with say blood, faeces and other organic matter you'd want to remove much of it before going to wash.
Most people today don't generate sort of heavily soiled wash compared to years ago. Unless someone works on a farm or otherwise is exposed to heavy dirt most laundry usually is made up of body soils with occasion stains caused by food, beverages or other substances.
Years ago virtually all European front loaders had a pre-wash as part of "Normal" or long cycle. Today pre-wash is usually separate or bundled as part of "Heavy" soil option.
For most domestic laundry detergents today (at least top shelf varieties) are able to remove and keep soils in suspension. This within limits of what testing considers "light, medium and heavy" soiling.