Strictly speaking, you are right to point that out. The .cat domain was established through the initiative of the puntCAT Foundation, a private entity whose Board (Patronato) includes institutions such as the Institut d’Estudis Catalans or the Corporacio Catalana de Radio i Televisio, among many others.
However, to be true to the spirit of the process, it must be noted that at the time the Catalan Government promoted, supported and disseminated the new .cat domain with at least as much enthusiasm as it has now put in opposing the .bcn one.
Not to speak of the direct links between the Generalitat and some of the institutional backers of the puntCAT Foundation. The Corporacio Catalana de Radio i Televisio, for instance, is a public body that manages media services for the Generalitat de Catalunya (as described in its own website).