Yes unfortunately I've observed this in some support systems. The best way is to thread the needle between being extremely personally polite to the other human on the line, but going through the required machinations on their runbook to trigger an escalation.

That is - you don't really have to behave unpleasant (raise voice, swear, be impolite, threaten) but you should just refuse to get off the line, demand escalation, and importantly emphasize with their predicament in needing to escalate you. Possibly including phrasing like "what do we need to do to resolve this issue".

I had a cellphone provider send me a $3000 bill because someone apparently was able to open 5 lines & new devices in my name/address. I went through the first few steps of their runbook including going to police department, getting report filed, and providing them the report number. They then tried to demand further work from me and I escalated.

At that point I turned it around - what evidence do you have that I opened this line. Show me the store security footage of me buying the phones, show me the scan of my drivers license, show me my social security number? Tim, are you saying I can just go to the store with your name & address and open 5 lines in your name? Being able to point out the asymmetry of evidence, unreasonableness of their demands, and putting the support staff in my shoes.. they relented and cleared the case.