The Teleparallel Equivalent to General Relativity (TEGR) is often presented as a gauge theory of translations, i.e., that uses only the translation group T4=(R4,+) as its gauge group. In a previous work we argued against this translation-only formalism on the basis of its mathematical shortcomings. We then provided an alternative proposal using a Cartan connection. Recently, a reply by some of the authors defending TEGR as a translation-only gauge theory discussed our objections. Here, we first clarify our arguments, and give new proofs of some statements, to answer to these discussions, maintaining our first claim. We then amend one of the arguments that originally led us to propose the Cartan connection in this context. This broadens the a priori possible choices for a TEGR connection.