Apart from Tulu and Malayalam regions, none of the other Dravidian regions had matrilineal traditions. This is indeed an intriguing situation and needs an explanation.
Sexual Divison of Labour:
There are studies that question the sexual division of labour in hunter-gatherer societies. Instead they opine that it's a phenomenon of sedentary societies. Considering that civilization first took roots in sedentary societies, we can be sure that it would have had patriarchal authority.
But civilization also rose in other hunter-gatherer societies because of contact with sedentary societies. I think the interplay between sedentary community and HG bands determined whether the resulting society became patrilineal or matrilineal.
Since there is no strict division of labour in hunter-gatherer bands, I would consider the family structure was matrilineal and there was no definite community structure or no authority.
Now the present day Dravidian tribes are also not matrilineal. To account for that, I propose that when the concept of marriage entered into these tribes, even though they remained as tribals, matriliny made way to patriliny. Patriliny is a conscious structure and identity whereas matriliny is a natural development which is a non-conscious identity.
But the question is whether it is possible for any matrilineal community to become a matriarchal society. I am going to rely upon Rhoda Halperin's view that food scarcity drives HG bands to adopt exclusive male hunters society. However, abundance would have both sexes in equal responsibility. Here I propose, only HG bands in food abundance regions have any chance of turning into matrilineal and matriarchal(or equal rights as there is no sexual division of labour) non-tribal society. However, we have to introduce one more variable into this. And that is the class of sedentary society that comes into contact with this HG band.
Matriliny and Traders:
I propose if priests and rulers meet the tribal society first then the HG band may turn into patriarchal one. However, if traders meet them first then there could be chances that HG band in food abundance region could turn into matriarchal or equal rights society.
In my opinion, the traders from eastern India moved to south-west coastal India in probably 4th or 5th century BCE. From there they sold black pepper to Arab and Greek regions and imported black-eyed peas. South India around that time was inhabited by numerous Dravidian tribes. And most likely Tulu-Malayalam region was a food abundance region and therefore without much sexual division of labour.
Be it patriarchal or matriarchal(equal rights), if the transition from tribal to non-tribal society is influenced by another sedentary society then it tries to build upon the structure of the sedentary society or legitimizes itself from the parent society. Here I try to explain the earliest known record of matriarchal kingdom of Pandyas. According to Megasthenes Pandyas were ruled by a queen who was the daughter of Hercules.
Here we can see two legends of north being adopted by matriarch ruling class to legitimize itself. The first one is the name Pandya. There are many speculations that Pandya could have been derived from Pandava. But why Pandava? It's mainly because there is a legend in Mahabharata that would legitimize female rule.
Arjuna, one of the Pandavas, goes to a kingdom ruled by women when the horse of Ashvamedha enters that territory. To cut the story short, the queen Pramila becomes Arjuna's consort or Arjuna becomes her consort. This story along with another Mahabharata hero Krishna (whom the Greeks identified with Hercules) was adopted by one of the matrilineal households after they took these traders as their consorts.
I would think this is how a strong matrilineal tradition was cemented in Tulu and Malayalam region. Maybe even Sri Lanka too was initially matrilineal/matiarchal. I suppose according to legends before King Vijaya, it was ruled by a queen called Kuveni.
Though these initial contacts with traders made the matrilineal tradition mainstream, as I argued before, the inherent weaknesses, which I would identify as non-conscious matrilineal identity, rendered it non-conscious slave of male overlordship. It is highly possible that one of the male heirs of Pandya line turned it into patrilineal structure and expanded into Tamil region*. Since inherently matriarchy is equal rights rulers need not be only females.
This could be seen in Malayala region where even matrilineal families were headed by only male members. However, the situation was bit different in Tulu region where I suppose the eldest female member of the family had the same rights as the male member.
*Here it should be noted that patrilineal Tamil Pandyas called their capital as Madurai after Krishna's birth place Mathura. We have already noticed that Megasthenes' matriarchal Pandyas also considered themselves Krishna's descendants.