Obviously it would be great if a schematic would lay itself out. Eagle and other applications have included Autorouters, that do just this, for years. In the case of Eagle this was a feature you had to pay for. Until recently, well recently in years. Of course the computer in this case can not do a better job than a human brain. That said it can do the job much faster.
Recently I decided to play with the Autorouter and see how much time it really saves. The feature is as easy to use as pressing a button. Pretty fast too. By default the autorouter wants to create a 2 layer board. If you are doing your boards by hand you’ll want to create a single layer board. In Autorouter Setup > General set 16 Bottom = N/A.
To set the trace size Go to Edit > Design Rules. In Sizes set the trace sizes. The default size is pretty small and will route traces between the pins of transistors. This might be a little tight with traces too thin for etching by hand.
I found the Autorouter to work best iterating by hand. Autroute your board. Take a look at the results and look for where you can optimize the results by moving parts around or changing their sizes. For example changing a resistor from a a .3″ span to a .2″ or .1″ span. Ripup the traces and Autoroute again.
Sometimes you run out of ideas, the Autorouter makes a nice collaborator, giving some good suggestions here and there. Sometimes the Autorouter will place an awkward trace snaking through the middle of your layout. By examining this you can often move some parts around and Autoroute again to get a better result.
Seems like a time saver. But I would add that iterating by routing, modifying and routing again can take time. So creating a board with the Autorouter is not at the speed of clicking a button, unless you’re happy with the first results. Getting the parts into a good position in the first place really speeds up the process.