Surely a circuit-board is just a way to hold and interconnect the components securely, and a breadboard is just a way to do the same while you experiment.
You don't need a huge custom-made circuit-board for a couple of resistors, an FET and a LED - a square inch of Veroboard would be enough, including the connection to the GPIO header.
But if you really don't want either, you could try bird's-nesting. Just solder everything in mid-air, and poor Araldite over the lot.
That tutorial is slightly alarmist, to my way of thinking.
The Pi's GPIOs are spec'd at 16mA, but all that means is that if you attempt to draw more, the voltage will drop.
Trying to feed a STP55NF06L, for example, which according to the pdf takes 30nC to charge its gate, would mean the FET would take about ½μs to fully switch on.
Putting a resistor in series with the gate would slow everything down, but keep the current below 16mA
That radioshack LED: it appears to have a resistor built-in, but there's no info as to what value it is, or the current or voltage the device is wanting.
But you could always build your own, with a LED and resistor inside a piece of tubing. You could even build an FET into it.