True about an underpowered soldering iron.. that is why it is better to get a good temperature controlled iron. I can easily solder most joints (except larger grounds) with my iron set to around 650-700°F.
The problem with cheaper irons especially on larger connections is that the heating element cannot react fast enough to the decrese in tip temperature causing the increased heating times. A cheap 15W iron will always put out the same 15W of power to heat the tip, reaching a certain temp. Once you start soldering, that temperature will drop and will not raise back up until you let the iron sit.
A temp controlled iron will always try to keep the tip at a set temp. It has been ages since I used a cheap iron.. but I know that they take forever to heat up. Imagine trying to solder with an iron that takes a few minutes to heat up to a low temp. A ceramic heater is much better and can heat the iron to 700 in 40 seconds.
This was interesting to actually see how low the temps really are even for me I'd go with at least a 25watt in that case for most general work but after seeing the vid, probably go with the 40W. Keep that tip temp hot enough
15W = 540° resting (soldering temp 450-500)
25W = 680° resting (soldering temp 620-670)
40W = 740° resting (soldering temp around 700)
It is when you get to some of the smaller passives like transistors etc that you have to be careful with the temperature you apply. Most datasheets will say what temp the leads can handles for a small amount of time during soldering.
**It is important to note that a good iron like a Weller will have a hotter temp with a given watt rating. (info from weller site, basic SL series pens not temp controlled so the temp will drop a bit like the radioshacks but not as bad.)
The SL325 is a 23W with tip temp of 700°
SL335 is a 33W 800°
SL345 is a 45W 900° iron.
The WP series are temp controlled.