Turkish Airlines International Travel Insurance and Flight Cancellation Issue

I purchased international travel insurance from Allianz Sigorta, opting for all coverages at a price significantly above the market rate. My insurance, covering the period from 22.12.2023 to 28.12.2023, was rendered useless when my flight on 23.12.2023 from Gaziantep to Istanbul to Batumi was canceled and rescheduled to 24.12.2023. (Meanwhile, Allianz Sigorta customer service only operates during weekdays, so I couldn't get proper information.) On 24.12.2023, I was issued boarding passes for both flights and sent to Istanbul Airport. Upon arrival, I was informed that the Batumi flight was canceled. No one from THY provided any information. Later, after my own efforts, they suggested sending me to Trabzon or Rize and then making my own way from there. Without offering any assistance, I was handed a ticket to Trabzon. After arriving in Trabzon, I had to pay out of pocket for a taxi to Batumi, which took 3 hours. Neither the insurance company nor THY is willing to reimburse me. Why did I even purchase the insurance then? The taxi driver didn't provide a receipt, and the place where I ate didn't give a receipt either. I am asking them to reimburse me based on the standard rate, but they refuse. It's a shame for these big companies.