In other words, Spaw, the penny whistle of the '30s?

The C melody sax was an "instrument for everybody" and was, without doubt, the easiest "keyed" wind instrument for people to pick up and play a little - and I'll agree, usually badly. While it doesn't require the tense embouchre needed for a clarinet in the "common" lower (2+) registers, when you get into the top 2 octaves (of the 4 octave range available) it very much requires embouchre control seldom developed by clarinetists (and almost never by casual saxophonists). Of course, using the upper registers of a C melody sax would sort of defeat the purpose of the instrument.

Most "professional" bands and "dance orchestras" used the Bb tenor and Eb alto saxophones. The Bb tenor could read from the Bb trumpet/cornet and clarinet parts easily, and the Eb alto could read the C trombone (written in bass clef, read it as treble and add 3 sharps to the signature). Much of the music written for these kinds of bands/orchestras was in "flat" keys, to suit the instruments used.

I can't be sure, but the original "Goofus" was probably notated in Ab. F, Bb, and Ab notations seem to prevail on sheet music I've "collected" from the era. (I'm not sure why Eb seems to have been less common.)

One of the "problems" with "melody" instruments is that one isn't ever enough. While guitarists "think in chords" the instruments that play one note at a time have to "think in harmonies." It takes a minimum of about 3 instruments to "get a sound" that's really musical, so the "solo" wind instrument player doesn't really have a chance.

Unfortunately, the C melody sax was marketed mainly to those who didn't have a band. Possibly the second most popular "home musician" instrument of the era was the mandolin (more frequently picked up by those with some music training); although one may also recall the "fads" with the zither and pan pipes from near eras.

Harry Chapin's rendition of Six String Orchestra will give you a more modern perspective on the "amateur musician" thing.

John