Yes. Fidesz won that supermajority with 52.73% of list votes, although they consistently polled more than that, in fact most of the time above 60%, from April 2007. That supermajority could have been even bigger, also because LMP rose above the 5% limit at the last moment. In fact I dug up an old poll from January 2010 (three months before the elections) asking about people's attitudes towards Fidesz's expected two-thirds majority, and a relative majority thought that it would be a good thing because at last one party could implement its policies instead of gridlock...

Even given Fidesz' nationalistic catholic rhetoric, was such a reform of the constitution in the cards?

Yes. Fidesz had this attitude towards rules and institutions from 1998, and did as they pleased in every field where they managed to get majority – parliament, parliamentary commissions, public media boards. They sabotaged a significant justice system reform. The majority ignored or forgot this, and when the Socialists warned against it during their campaign, they did so with zero credibility.

The complete replacement of the 1949 constitution itself with a new coherent text was something put on hold in 1989 with a sense that it should happen sooner or later (the preable added in 1989 explicity says that the amended old constitution is transitionary). However, during successive legislature periods there ws always too much acrymony between the political sides for initiatives not to falter for lack of cross-party support. There is a little sub-story here: the 1994 Socialist-liberal government had a two-thirds majority, too, but they thought that as a government with an ex-communist main partner they should demonstrate a lack of hegemonic aspirations and adopted a two-thirds law that a new constitution needs a four-fifths hiper-majority. One of the first things Fidesz did was to squash this law. Although at the protest, speakers made a point about Fidesz not having made a constitutional change a campaign theme in 2010, I did remember them speaking about it, and with a little search found that Orbán railed againste the untenable 'Stalinist' constitution that is a 'technocratic jumble of laws' in November 2009.

_{*Lunatic*, n.
One whose delusions are out of fashion.}