So there is something else restricting the voice, which is usually self-consciousness. They way to free up the voice is to be able to sing in an environment where nobody is judging you. I clearly remember at University having to play the violin in front of my class, and before my tutor who was viscious in his criticism of pitch and tone. As a result, I would sweat, clam up and fulfil his own prophesy.
We need to perform in an atmosphere of joy and generousity. Criticism is of course vital to improve, but it can be given in a non-judgemental way. The cliche of criticism being given between praise might seem overused, but it still works.
So when singing an improvised song, it is crucial that the self-critic is turned all the way down. What would we rather see, someone singing a perfectly rhyming funny verse with no commitment, or someone singing with joyful abandonment? Musical improv does encourage people to just go for it. There is no substitute for opening up, connecting with the audience and just giving it your best shot.
It is the same in non-musical improv. When required to do an accent, it is far better to give it your best shot, give a look to the audience and then keep going with commitment, than it is to half attempt it and then try to drop it without anybody noticing.
I guess if workshops are rehearsals for shows, then shows are rehearsal for life. It is a chance to explore and fail without consequences, without criticism and without judgement. When those situations come up in real life, maybe we are more equipped to deal with them - I mean how many times have you had to pretend to be a traffic warden with a Brummie accent? Exactly.