I think the use of a word conservative here is interesting to say the least. I'm used to saying it's the strongest assumption, the one that's hardest to prove that it holds and frankly the one that's probably violated most easily.
It's the assumption that's easiest to build upon when teaching the regression theory. You don't need to worry about correlations and all the problems that they bring. You can easily apply CLT to get the asymptotic variances of parameters etc.
You'll notice how easy it is to work with i.i.d. errors the moment you start talking about time series. All of a sudden you realize that the assumptions that are somewhat reasonable in cross-sectional analysis, do not hold in time series usually. Even in the cross-sectional analysis you don't really need independence and get get away with weakened assumption, e.g. see Gauss-Markov theorem.
To me semantically it's better to use a word conservative when referencing the weakest assumption, i.e. the one that should hold true in most situations, not the strongest one, that holds rarely if ever. I would call i.i.d. assumption the most liberal, because it also liberates you from the necessity to deal with all the correlation and dependence issues, it lets you build this wonderful ideal world of independent errors. I could also call IID assumption outlandish