What exactly were TETRA radios used for? I assume they were government/infra related, but then I don't understand why they'd need to backdoor the keying
They are used for many things, like fire, ambulance, railways, harbour operations, police, military, coast guard, and so on.
The weaker cipher mode, TEA1, is used when selling the radios to anyone who may not necessarily be an ally or highly trusted. This is the legacy of strong crypto being export-controlled.
It was public that these ciphers were weaker, but they were actually much weaker than advertised. This is the backdoor.
They don't so much backdoor the keying as that they have 4 different cipher profiles, and the one approved for global rather than European use (TEA1) compresses the key from 80 to 32 bits.
It's essentially a surreptitious version of what the US did in the 1990s with "export ciphers".
And that is supported by the known past actions of "some government authorities". This is definitely not the first time the US government has deliberately sabotaged crypto.
I think the most relevant use in the context of deliberate backdoor is its use by police and military forces. Apparently some energy providers also use it for remote controlling tasks (no voice).
The weaker cipher mode, TEA1, is used when selling the radios to anyone who may not necessarily be an ally or highly trusted. This is the legacy of strong crypto being export-controlled.
It was public that these ciphers were weaker, but they were actually much weaker than advertised. This is the backdoor.