Aircraft used to fly rhumb line tracks in the days before they had global navigation systems like FMS, IRS/INS and GPS linked in to the autopilot. They had to do this because their heading reference was either a DI or a compass or a combination of the two and to fly a great circle track, the shortest route, would require a constantly changing track direction which would be extremely hard to manage manually. You still do this today for shorter legs in light aircraft, you draw a line on the chart and either measure track in the middle or measure it at both ends and take the average. The difference isn't much but by doing that you are flying the rhumb line, which is approximately half way between the initial and final great circle track.
In the old days long tracks were achieved by approximating the great circle track, which in its simplest way involved a rubber band on the globe as described above, then flying a series of rhumb line tracks between waypoints to approximate to the great circle. In the example you quote for LA to London the great circle track would take you near Greenland, and you would fly a series of airways, direct tracks and NAT tracks to approximate to it, either flying rhumb lines with old navigation kit or great circles with modern kit. The single rhumb line track would go much further south, and never go north of London's latitude, but you would never fly a single rhumb line track of that length, the route has always, even in the earliest days of navigation, been broken down into a series of shorter tracks from waypoint to waypoint so that you can assess your progress and tracking.