> If I wanted to go with a bit more traditional power supply, using a transformer
Living in the past.
Assuming full 20W output, the raw DC should be around 11, 12, 14V with at least 2 Amperes. OTOH at idle current goes near-zero yet voltage must not rise much over (IIRC) 18V.
That's a very tight spec for a small transformer. Only 28% different, and small PTs have 20%-25% regulation, plus 5% or more uncertainty in wall-voltage, plus you can't get the eXacT voltage rating you want without custom-costs.
18V-14V 2A DC means a "10V or 12V 3A" transformer, a 10A FWB, and about 10,000uFd of 20V cap. And a line-fuse! Depending where you shop, that's $20-$40 of heavy metal.
Plus the extra-care needed to wire wall-voltage without shock or fire. You can do it; it is much better if some other person checks your work (we are most blind to our own oversights).
OTOH apparently excess flatop supplies sell REALLY cheap. The wall-voltage safety issues have been DEALT-with, in design, testing, certification, and by the fact that they made thousands of them before yours (not a "one-off"). And the supply is regulated just a little above the voltage we need, so that a not-excessive $2 regulator can nail the supply voltage for-sure.
And yeah, if you don't like cord-bricks, tuck it inside. These bricks do need cooling, but for the typical flaptop brick and this amp they don't really need to be out in the open.