As you said, the Directory wasn't elected and they had the executive power. The Directory itself was installed by a coup. They were in no shape or form a democracy.
That's not really comparable to a prime minister elected in a two-stage election like the UK.
They were elected by the Council of Ancients (upper house of the national legislature, which was itself elected) from a list sent by the Council of 500 (lower house of the national legislature, also elected.)
> The Directory itself was installed by a coup.
No, it wasn't. After the Constitution of Year III was adopted, the two councils were elected under it and then the Directory elected by the councils.
The Directory was a multimember executive of a nominal democracy, so, no, the upthread description was entirely accurate.
(The Directory wasn't itself directly elected, but then neither are a number of executives, individual or group, in democratic republics.)