Your mean of quartiles is very much like Tukey's trimean, which is (LQ+2M+UQ)/4 rather than (LQ+M+UQ)/3. I expect it has broadly similar statistical properties. Tukey was a smart chap and I would guess the trimean is "better" for most purposes. But, as Lumifer says, what counts as better will depend on what you're trying to do and why.
(E.g., the trimean will have a better breakdown point but be less efficient than the mean; a worse breakdown point but more efficient than the median.)
(E.g., the trimean will have a better breakdown point but be less efficient than the mean; a worse breakdown point but more efficient than the median.)
What does "efficient" mean, in this context? Time to calculate would be my first guess, but the median should be faster to calculate than the trimean.
In a previous post, I looked at some of the properties of using the median rather than the mean.
Inspired by Househalter's comment, it seems we might be able to take a compromise between median and mean. It seems to me that simply taking the mean of the lower quartile, median, and upper quartile would also have the nice features I described, and would likely be closer to the mean.
Furthermore, there's no reason to stop there. We can take the mean of the n-1 n-quantiles.
Two questions:
Note the unlike the median approach, for large enough n, this maximiser will pay Pascal's mugger.