Small class sizes are literally the biggest expense though - teacher salaries dwarf everything else in a school, even flashy facilities.
The actual data on academic achievement mostly finds that small class sizes don't improve academic outcomes unless there are special educational needs.
So if we want "budget private schools," the best model would be "private secondary schools in big cities that have pleasant but basic facilities, quite large classes and a basic but not extreme entrance test," on the grounds that children who are at least average in terms of academics and behavior can cope fine in normally sized or biggish classes and this would make running a school an awful lot cheaper.
My daughter goes to a private secondary school outside the UK that is similar to this, but my experience is that UK parents tend to really like small class sizes, so not sure this model would be a big seller in the UK unless the local state secondaries are pretty bad.