Yes - anyone can sing and an already competent singer can improve -but only if they are given the tools with which to match the pitch of the sound being produced with what the ears are hearing. In my experience the so-called 'tone-deaf' actually have very acute and accurate hearing capability. Out of my scientific training and my instrumental experience (specifically trumpet and natural trumpet - fitting well with the subject of the article under discussion)(see website for trrumpet connection. I have devised and produced a voice development programme - 'Calisthenics for the Singing Voice' - The programme is documented at <http://www.voicebuilder.co.uk> Results achieved by my use of the pogramme documented at <http://www.voicebuilder.co.uk/case_studies/case_studies1.htm>
The following quotation describing the modus operandi of the Royans Institute for non-surgical voice repair illustrates very neatly my own approach in working with damaged voices (see case studies above) as I devised the exercises which are all designed to lift the voice off the voice-box (and keep it off the voice-box).
quote< We will have to employ our revolutionary approach to voice mechanics by lifting your voice off of your vocal box and restructuring it into your set of facial muscles and then putting those facial muscles to work in full conjunction and coordination with your abdominal muscles, thus, minimizing the use of your throat, larynx and vocal chords. While you are not using your vocal box excessively and practically not all for speaking or singing, your vocal box will be healing simultaneously with the application of the natural herbs and remedies. >unquote - Quoted from Singing Teachers/coaches - Importance of Peripheral Thinking for singers. ( see next comment)
My system is based entirely on the exercises but I can see that natural remedies would also assist with physical damage -(My subjects were all struggling to overcome nature's unkindness) examples of the exercises can be seen and heard at the following links. <https://youtu.be/Lvw6_NhyRVQ> <https://youtu.be/Bc7CvkuRynE> <https://youtu.be/7AY-vL9eiek>
These presentations are in one pitch range for medium voice - In the full programme there are 6 overlapping pitch ranges for each exercise facilitating exploration of the whole possible range of the voice.