Beware of Private insurance players

I bought a second hand car recently. The insurance was covered by Reliance ADA group. I was busy with certain domestic functions and was travelling due to which the renewal of the insurance skipped my mind.  Neither did the insurance got in touch with me for renewal.  When I contacted the insurance company, they insisted that I renew the insurance comprehensively as otherwise they have instructions from their higher ups not to renew it.  When I asked them a simple question that the company did not even bother to send me a reminder or get in touch with me for renewal, how could I trust the company for claims that may arise in the future?  They sales manager Mr Jayaraj who works for Anna Nagar Branch that issued the policy, expressed his inability to renew the car insurance as third party!  When I asked him to give it in writing, he simply refused to do so and said it was not company's policy to renew the existing policy into a third party policy!  This argument has no base and does not have the support of IRDA, the authority which controls the insurance players and can act like an Ombudsman.  I contacted New India Assurance company Ltd who are my neighbours on G N Chetty Road, T Nagar, Chennai.  The officer politely refused to renew the insurance cover saying that they have instructions not to accept third party renewals?  This appears to be a joke as the statutory law does not permit to drive the car without a valid insurance and no insurance company is willing to renew the policy as third party!  What should I do is the next question?  I took up the matter with the head office of the New India assurance and also the Reliance ADA.  While the e mail sent to the government owned insurance company got returned undelivered, the Reliance ADA group has a toll free number who are enthusiastic to accept comprehensive renewal but not third party.  The reason for my insistence to renew the policy as third party is because of the experience and fact that the insurance companies do not pay any heed for accident claims and keep you waiting and eventually allow only 1/3rd of your claim stating so many reasons for disallowing the claim.  They do not pay anything for rubber or plastic parts and electrical lights and ask you to pay upfront for repairs and reimburse you a much reduced amount after deductions etc..
The third party insurance is useful if someone is hurt or run over by your car due to accidents which is rare as you drive to reach safe and not to hurt anyone?
I am still driving the car without an insurance cover with a hope that one of these companies will renew my policy as third party soon?
For a country like ours, too much freedom and free market economy does not work or deliver.  Our country is based on socialistic principles and things will be better if controlled by government and not private entrepreneurs!  The government run by the late PM Indira Gandhi is the best module suited for our country.