The following is what I gathered from two musician-friends on 'Hamsa' connection to Hamsadhwani raga.
Hamsam looks clean and is therefore a symbol of purity
It is white in colour and therefore a symbol of sudhdha satvam
That is why those steeped in suddha sattwa are known as Parama hamsas
-eg -Raamakrishna paramahamsa.
Kavi Sadasiva brahmendram placed a 'parama hamsa mudhirai' in his krithis (15 or so)
About the dhwani - the sound or the talk of swan:-
Even though the sound of the swan is unpleasant (karNa katooram) at times,
it has emanated from sudhdham – the clean or pure looking swan.
The hearer has to prepare himself to be ready to receive that and make himself assimilate the pure and sathwa sounds / talks given by a hamsa or Parama hamsa.
Hamsadhwani is a little harsh or unsmooth to begin with (like the sound of the swan).
This can be understood by comparing it with Mohana raga.
These two ragas differ only in one swara.
The 'Ni' of Hamsadhwani is replaced by 'Da' in Mohana.
With this one change, Mohana gives more 'sukha' right from the first swara than Hamsadhwani.
In contrast, Hamsadhwani starts with something like that of a warming-up exercise so that the mind is well prepared to receive the finest notes of the raga later, that gives the finest enjoyment subsequently.
This happens how it is experienced in hearing the sounds of swan.
The sound of swan is unpleasant, to say the least. But when one looks at the swan that gives out the sound, its appearance is so overwhelming that one starts watching it while hearing it. The experience of watching the soft and pure looking swan transforms the senses in such a way that one starts to enjoy the sound it makes! Whatever unpleasantness or stress that one may be having, is gone when one watches the swan 'singing'.
This rationale is applied in Ganapathy songs sung in Hamsadhwani raga.
Lord Ganapathy is the destroyer of obstacles.
The obstacles or 'vignams' coming in the way of singer, say,
the saareera vignams such as sudden trouble in voice, trouble with sruthi and in present day situation. trouble with mike and sound systems would be gone, similar to how the unpleasant notes of Hamsa are forgotten while seeing the Hamsa itself.
That is why most singers opt for Hamsadhwani raga to praise Lord Ganesha in the beginning of a concert.
This raga was first given by Ramaswami Dheekshithar, father of Mutthuswami Dheekshithar (one of the tri-moorthis of Carnatic music).
The famous song on "Vathaapi Ganapathy" was given by Mutthuswami Dheekshithar in Hamsadhwani!