When there's an obvious form of dependence to check for, it usually should be, and often is.
For example if data are observed over time, there's an obvious kind of dependence to check for, and obvious ways to check for it (e.g. time series plot, ACF/PACF). Similarly with observations related spatially there are obvious forms of dependence to check for (e.g. that neighboring observations might tend to be related) and common models for such dependence which suggest useful diagnostics.
The problem is that in general there's usually no obvious form of dependence to check for, and a huge number of ways in which there might be some kind of dependence (but likely isn't big enough to worry about). Of course if there's some blocking factor or likely random effect, or known potential covariate, that's an obvious source of dependence, but often in practice people simply fit a different model for such cases to deal with the issue; if we assume that's not the case (because we're still fitting a one-way ANOVA), the question becomes "what dependences do you want to check for?"
A small aside on an assertion in your question:
The standard procedure for a one-way analysis of variance seems to be:
1. fractile-diagram (or Q-Q plot) for checking normality
2. Bartlet's test for heteroscedasticity
I find it curious to use a plot for one and a formal test the other. I believe more usually people tend to test both or use a diagnostic plot for both.
While I think a Q-Q plot is fine (though in many cases overly worried about), I wouldn't formally test heteroskedasticity, since it answers the wrong question (the relevant question is about effect of the hetero, not its existence; diagnostic displays are better for that)*. And if for some reason I did test it, I wouldn't use Bartlett's test, since it's sensitive to even fairly mild non-normality.
* if you then choose a different procedure when you reject it also adversely impacts the properties of both the ANOVA and whatever else you did. Generally it's better to simply not assume homoskedasticity to start with.
There are numerous threads on this issue on site; some reference papers that recommend against any formal test of heteroskedasticity.