Insurance policy is just a piece of paper - Insurance company actually sell 'claim' - i.e a buyer wants hassle free claim processing. For market giants like Adamjee and EFU, their policies/clients are accepted with much ease by all dealerships along with local denters/vendors. secondly, the claim settlement is very swift - no questions asked if the loss incident in genuine and all required info+documents are provided by client. also, multiple claims can be taken in the tenure and policy doesnt get exhausted once the 1st claim is lodged.
it is very much seen that smaller companies insure cars at 'marginally' lower premiums and when it comes to claims, it is pure hassle for client when they start to 'allow' and 'disallow' repairs/replacements etc.
so as per my whole insurance industry's experience, it is always beneficial to insure with big names as the premium is not very high. like Adamjee is just 2.5% and EFU is same too or a bit higher.