Maybe I'm looking at this wrong. I agree with everyone above
Melody on its own isn't enough. However, to take golf as an analogy, the ability to drive off the tee isn't enough, either. To play a good round, lots of other skills come into play (unless you always get a hole in one!). It seems to me that, if driving is your weakness, practicing your driving skills would be a good idea. In the same way, I think it might be helpful to develop my melody awareness (and I hope that some of you better melodists might share your approaches - if you can (it might just come subconsciously)).
Simon is right. When I listen to someone singing a song I know (or if I sing it) a capella, I do "hear" some of the surrounding instrumentation. Of course, this isn't true of songs I don't know. With songs that I don't know, there often needs to be something else to hold my interest. The singer may sing/tap/clap a bit of percussion, for example. Often, it's the lyric. One genre that springs to mind where an unaccompanied singer is fairly normal is folk. In folk, the story is pretty key (though it can be the simple beauty of a lament, but there is a story of sorts there too). The lyric holds the melody up and keeps my interest.
There are exceptions. Take a tenor singing something like Nessun Dorma. I may not understand the words, but the melody is enough to keep my interest (well, that and wondering if the tenor is going to make every note!

).
Then again, rhythm is part of melody. If you sing "Anything Goes" really slowly, the melody is pretty boring (I think). Speed it up to it's normal level and I think it's a cracking melody!
I suppose I am agreeing that the song is made up of lots of interlocking devices, all of which are important. Is a "strong" melody vital? Probably not. Does a more "interesting" melody always improve a song? I don't think so. The core of many songs isn't the melody, and the "feel" of the song could be destroyed if it soared and swooped more.
One could argue that the phrasing is more important. We have all heard songs that are fabulous when one person sings them and awful when another destroys them (I'm thinking of some of the covers done by more recent artists of old hits, where they have sometimes destoyed the soul of the song with their vocal gymnastics - think Mariah Carey).
However .. however .. I would like to be able to produce more interesting melodies when I choose to. I suspect that I am constrained by bad habits rather than by choice. I'd like to practice my driving!
How to do it is a question. I wonder if we are nervous about putting up a melody on its own, fearing that it won't be sufficient on its own. Should we la-la it or play it or sing it with the lyric? I have to say, I would find a melody with no words hard to comment on, in much the same way that Simon finds it hard to consider a melody without the context of the backing.
Does anyone here usually write the melody first? If so, how do you go about it? What is the process of developing a melody, and then the other parts of the song? As I write on guitar, I think I try and find some sort of back and rhythm and melody all at the same time (I play something and sing over it.. la-la-la).. and then words come. I'm sire there are other ways - maybe easier with a keyboard?