John W. Tukey called this the trimean. (That term is also used in the paper you cite.) It features more in the preliminary editions of Exploratory Data Analysis (three volumes, 1970-1971) than in the fully public 1977 version (both Reading, MA: Addison-Wesley). (More is a little ironic here: I am not sure it is mentioned at all in the 1977 book, but do not have a copy at hand.)
The context for Tukey, his students and his readers around 1970 was identifying median and quartiles by hand in a batch of numbers (usually as individual quantiles or order statistics or averages of adjacent quantiles) and then -- in the case of trimeans -- combining them with weights. Although Tukey was involved with computers from the late 1940s (suggesting the word bit and being an early use of the word software and suggesting hardware tweaks too), his exploratory style around 1970 put very heavy emphasis on mental arithmetic with paper, pens and pencils, both for himself (he did a lot of exploratory work on planes and in committee meetings) and for others. He was keen on as far as possible just ordering, counting and averaging pairs of numbers. The technological context was that, if I recall correctly, cheap portable calculators were not routine until the late 1970s. Use of computers was a long way from the obvious everyday and universal practice it later became.
Similar ad hoc devices were earlier in routine use in sedimentology. !!!Details to come.
All that is history. A detail that may still bite occasionally is that if all you see in a report -- especially of other people's work -- is say median and quartiles, then a trimean combines the information in all three, uses more information than the median alone, and remains more robust than the mean. The weights 1/4, 1/2, 1/4 are arbitrary other than summing to 1, unless someone can suggest a rationale. In Tukey's case cutting down on arithmetic would have been a strong incentive for those weights. There is also an echo of weights (1/4, 1/2, 1/4) for moving averages, which Tukey called Hanning, although Julius von Hann surely didn't invent it.
I'd say there is a continuing rationale for trimmed means, but the usual context now is having all the data in memory. Averaging all the values between the quartiles is a 25% trimmed mean, in use since the early 20th century at least and often called a midmean. People familiar with box plots can think of this as averaging all the values inside the box. There is small print about the exact rules, which can differ. I'd tend to use midmeans rather than the trimean. Indeed, seeing how trimmed means vary with the trimming fraction is a simple tactic, discussed by (but certainly not invented in) this paper.