If private school parents/grandparents were honest we can all see these trends. For years there has been a move towards co-educational from single sex; to all-through schools from separate prep/senior schools; and from boarding to day schools. I'm talking about the 'ordinary' private school sector here- not the very top end.
DN was in a private school that closed twenty years ago- it was all-girls and had dwindled to about 60 pupils, coincidentally the Cathedral School in the same city had recently moved to being co-educational, and about half the girls including DN ended up there the following year.
The private primary that DGS was attending has closed since the VAT announcement and listed it as a contributory factor. The reality is that school rolls have been trending downwards for years, and there were 6 children in his year, which just wasn't a sustainable number without cranking the fees to a level the school couldn't justify. The biggest factors- the local state primary has steadily improved, eroding the differential; and the senior schools this school once fed in the area have mostly moved to a 5-18 model or have their own primary schools now, and offer facilities you would previously only have had post 11.
As someone said up thread, these schools are businesses even when run as charities, and they absolutely compete for both quality and quantity of pupils.