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This one has a pretty quick answer but is a great question as it relates to my IHS presentation next week and is something for sure people wonder about when considering a single B-flat horn. There are two main elements to the answer.
First, the F extension is really handy for low notes. If you have a four valve single B-flat with a stopping valve the lowest note you can play down to chromatically is low B-flat and then you have a gap until you can produce a low F. Sometimes you need those low notes! Some of them will be somewhat out of tune, the valve slides are not long enough when used in combination with the F extension (for example a low A fingered 12 and F extension will be quite sharp), but at least you have something to work with.
A secondary thing it does is beef up the sound. It certainly leaves a single B-flat sounding more like a double horn, as part of what makes the sound the way we are used to relates to weight of the instrument.
So actually Dennis Brain had the best of both worlds in a way, he looks to have only added the F extension to the Alexander single B-flat that he used primarily late in his career when needed for specific notes in specific works.
In the photo here my F extension may be seen; it is plugged in at the stop valve of the horn featured in this article.
It would be a problem to play stopped notes on this horn with the F extension in this way, which is why players who were strongly single B-flat players back in the day often used five valve versions with a stop valve and an extension.
Then you could ask, will or is the single B-flat coming back? Maybe—it is possible anyway. If you have never tried one and are fluent with B-flat horn fingerings it is really worth trying one, it certainly has a place and has a different, more responsive feel than the typical double horn or triple or even descant.
