The principal thing that took time at Salisbury, if a locomotive was to work through from London to Exeter, was taking water. Unlike the other three main line companies the Southern did not have any water troughs, and from earlier times the LSWR had built "water cart" high capacity 8-wheeled tenders. But by the time these got to Salisbury it would need a complete refill. There might even have been an associated issue with the water pressure there which meant it just took too long. And of course, if you optimise the tender for water capacity, there is less space for coal. Better and quicker to change engines. You were changing crews anyway.
The water issue was, indirectly, responsible for the 1906 Salisbury accident. The LSWR wanted to run boat trains from Plymouth to London with just one loco change. Doing so at one of the traditional change points, Exeter or Salisbury, meant that one leg or the other was just too long for water, so they ran the Salisbury loco 25 miles west light engine to Templecombe, changed locos there, and could then achieve it with two long runs but within the loco's capabilities. It then became about the only express that passed through Salisbury non stop.
The Schools fleet was addressing a similar infrastructure issue. The Southern's constituents had large fleets of existing 4-4-0s, and they just didn't have the turntables on the system to turn a larger loco, so a more compact but powerful one was needed at a range of points. The summer Waterloo to Lymington services needed to run a Schools for a long distance down the SW main line, just because there was no adequate turntable anywhere near Lymington for anything larger.
