Might it be worth looking into whether it's legal to record phone calls where you live? Some places all you need is the knowledge of 1 party (so, yourself) and then you have proof of whatever run-around they're giving you. Some places need both parties to know though so definitely be careful with this. Though this can be a bit complex when it comes to showing a third party, so definitely if you think you want to do this look into it more / possibly consult legal help.

I don't think you should default to having it come out of your own pocket if you paid out for insurance AND have all the receipts! Gather evidence and escalate it until you get your money back. It's not blackmailing, it's enforcing the service you paid for (insurance) which is essentially a contract: 'pay us x agreed amount and if something goes wrong within y specifications, we will reimburse you'. Or at least that's how all insurance I've ever had worked like. They will try anything not to pay out for something that expensive though.

Do you have the paperwork still on what they specifically cover? Or was it like, an online ToS type thing?