Do singers need to practice singing without accompaniment? If so, should the singer always practice singing the melody acapella?
There are times when I need to sing with no accompaniment in order to accomplish what might be referred to as muscle memory for my voice. When I repeat sections of a song over and over and can hear precisely what my voice is doing, it makes it possible for me to alter expression and emotion, play with the rhythm and check my pitch. I can best do this without accompaniment. When I have it worked out to suit myself, that's when I work out the accompaniment to compliment my singing. All the repetition helps the singing become a little more automatic and still communicate the emotion I wish to convey. However I won't say that my technique will work for everybody. Folks need to find what works for themselves.
Reading between the lines of this question - are we asking if singers should practice singing without the original version playing? A lot of people 'sing along' with a recording. Take away the recorded voice, they realise it was doing all the work!
If a song is GOING to be accompanied, by live piano or backing track, there's no particular virtue in singing it without. But you should certainly be able to sing it without the support of the original artiste!
On a more general level, not about learning one particular song but about learning to SING, yes, ability to sing in tune (and in time) without any support is very important. But your teacher must check that you ARE singing correctly. It's very easy to mess up the timing, or even to sing each phrase in a different key in order to keep in your 'easy' range.
I would say: Yes, they need to! But they should not always. The aim is to be independent of the accompaniment but also to sing according and listening to other instruments.
But like Dekkadeci seems to suggest in his comment:
This will be a process. It is not helpful if they can't keep pitch and get out of tune. The best is you start with playback singing and then always turn the playback lower.*1) It will also benefit to accompany yourself by a keyboard or a guitar ... and then training to sing one verse with and one verse without accompaniment to have control of the pitch.
If you have a bad result you could also check after a phrase with a tune fork, a recorder, a keyboard if you are still in pitch and you will find out the measure or the interval where you get out of pitch - often it is not the ear but the voice technic e.g. a part is too high.
*1) What we used to practice at school with our classes was so called "tunnel singing":
The children sing with support of play-back ... and suddenly I turn the volume down: this is a funny and joyful practice to keep the pitch and also to stay in time.
If you want to increase your ability you may train to sing a tune while there is another playing in a different key. Or study a song of Webern while you are listening to a popsong or an cantata by Bach.
The answer is: "both."
You need to be able to sing on your own - see "Solfege" for starters -- to train your voice and your brain to be able to stay on pitch.
At the same time, you need to learn to sing with accompaniment, be it vocal or instrumental, so that you can stay in tune with the group even if they go slightly off-pitch. It's always better for the group to be in tune with itself than for one person to be right on-pitch but out of tune.