Question: For example, last year I was looking at Bogota, Colombia to LON. This was quite pricey, but it turned out if I bought separate tickets - Bogota to Orlando, Florida, and then an Air Berlin flight to London was pretty well priced.

Even better, I checked and it went via Dusseldorf. However if I just bought the leg to Dusseldorf, caught the 20 min train to Cologne and flew from there, it got cheaper still.

This of course requires hours of messing around with flight websites, and often knowledge (like that Dusseldorf and Cologne are close by). Surely there's a simpler way?

Eg, I'm looking at flights from Vancouver to Bangkok, and also from LON to either SCL or EZE or Lima or Quito. One to Quito goes to the Netherlands, somewhere in the Antilles islands, onwards to Guayaquil in Ecuador and then to Quito. 4 flights requires all sorts of playing to find the cheap legs, if any. Surely there's a 'cheapest paths' search engine that can take care of this somehow, or a simpler way?

Answer: 
For multi-city trips, unfortunately, I don't think an 'optimized' search engine exists, and the sites like Kayak, mostly route using a single network of airlines, which may or may not be the cheapest option. 
You do have to spend time and be a bit creative in charting itineraries. Here's a protocol that has worked well for me -

1. Chart out the major cities in your itinerary, and work out the most efficient routes connecting them. The trick here is to factor in as many round trips as your time permits - needless to say, round trips are significantly cheaper than one-way options. You may have to find a central destination, and establish that as a hub - a well-connected airport with an international airline hub will serve well.
2. For the smaller cities, look up regional, low-cost airlines (the likes of Vueling, Southwest, etc.).
3. Finally, connect all the dots.

This requires effort, but in essence, follows that same 'hub' concept that has existed, in the US at least, for many decades.

Good luck.
Expedia and lastminute flight searches will show you flights with stops. It also shows the Cheapest flights with 0, 1 or 2+ stops for you to compare.

While it might not be as cheap as booking all the flights individually yourself (it seems fairly cheap £610 from vancouver to bangkok next month), expedia does make sure your carriers are in the same airline group, meaning that if your first flight is late and you miss your connecting flight, your carrier will get you on another one. If you book your flights individually, you will not get this protection (you can claim on your travel insurance, but i'm sure there will be an excess)

Also if you buy each ticket individually you probably have to go through immigration and re-check in at each stop.