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There's probably a few reasons behind the poor service. It's a badly paid role with high turnover so difficult to retain well trained staff. Plus there's not actually many of them because of the push to digitise everything and because the civil service is apparently far too bloated as it is...

As for them being sacked if it was privately run well more likely it would be fully outsourced instead and the customer experience isn't exactly great for services which do that either. It's just symptomatic of a general trend where customer service is not a role which is well resourced anymore or characterised by actual human interaction. Appreciate it is very frustrating though

I never actually worked in tax but I did work for HMRC for a while. Sadly no longer though so I can't even pull any strings on your behalf