The merit in your argument is flawed. Why do you think are the British and the Americans or the Australians close allies? Because they have respected and tried to support the diaspora of their original population whenever needed.
The fact that the Lanka India relations are stressed due to the Tamil cause is correct. However, please understand that it is not a handicap that there are Tamils in Lanka. Actually this is an asset. India does not need to fight with Lanka for our Tamils but at the same time also not abandon the Tamils in Lanka. If Lanka wants to hold the Chinese card in India's face then no need to get so much flustered. Any country in the world can do it. Long term diplomatic relations are not based on mollycoddling. Neither is it a case of being the highest bidder. If the Lankans want bidders then better not participate. However, instead India should evaluate these matters economically. China nevers gives a free lunch. Neither should India. It there is going to be some sabre rattling then why not do so. The maritime boundary renegotiation will be a pretty good starting point. We need to suppor the Sri Lankan Tamils to come to the mainstream in Lanka over the period. That is the message that the Indian Tamils need. The histrionics aside, even the Tamils in India need that comforting approach from their country.
You can thank India for Singapore too. One look at your culture, your langauge and your king should be enough to evoke those feelings.
You are correct, however the histrionics of the Dravidian parties is certanily not helping the Tamil cause too. This is more for local consumption and has no statesman like merit.