A few problems with the past statements:
The visible spectrum is what we can see with our eyes, therefore the limitation of the measurement in this case is our eyes. You are correct in saying that our eyes are 'restricted' to a certain part of the spectrum, but just word it in a different way, such as 'since our eyes can only detect a certain range of wavelengths..."
Secondly, I am fairly certain vapourised metal atoms will emit light, since the electronic transition at that point is quite low (don't quote me on it, but I am pretty sure that's the case). So I wouldn't say a flame is insufficient energy-wise. You would be more worried about light being emitted in the infrared and UV region that can't be seen by the human eye.
Also, another main problem with the 'distinct colour' is that two metals can emit different colours that are similar, but our eyes can't tell them apart. And also, the chemistry of metal cations in flame is not very well understood, and sometimes reactions can go unexpected depending on the composition of the flame/fuel/etc.