The Austrian Empire was an incredibly diverse country and a substantial part of its population was non-German, according to the 1911 k.u.k. census only 23.36% (about 12 million) of the population reported speaking mainly German. Though keep in mind that question was about usage and not mother tongue, so a Bohemian immigrant in Vienna could still count towards the ones reportedly speaking mainly German. Followed by 19.57% Hungarians and 12.54% Czechs.

In the pre-WWI 1900s the Habsburg capital of Vienna had the second-largest Czech population after Prague and beginning in the 1850s with the sharp increase in industrialisation Vienna's Bohemian population always remained around 20-25% of the total, in the last census of Vienna in 1900 it was 24.5%. And even though Bohemia had a sizable German population itself (Sudetendeutsche) it was only ethnic Czechs that emigrated in great amounts and mostly filled proletarian roles. Maids for example were a nigh-exclusive Czech occupation in that regard. Still, 46.4% of the city's population in 1900 reported being born there so were at least second-generation immigrants and 15% came from the Austrian crown lands which mostly had a German majority as well.
I suppose that is the source for Hitler's claim about the Slavification (?) of Austria, even if it's sort of stupid.

Now concerning the deadly enemy of Germanism that would be mostly right from Hitler's point of view in that the Archduke was openly striving for a solution to placate the Slavic population, which then either had to sort-of suffer under Austrian rule or properly suffer under the Hungarians, who forced a rather strict policy of Magyarisation compared to the Austrians' more lenient treatment of their Slavic population. Franz Ferdinand's goal was to give the Slavs what his uncle had given the Hungarians in the Ausgleich of 1867, a mostly-independent state to themselves. This concept of Trialism would have united the crown lands of Bosnia, Croatia and Dalmatia into Südslawien ("South Slavia") with its own parliament and everything.
Although his preference rather quickly turned towards a federal system, such as the one suggested by Aurel Popovic, which would have meant a complete overhaul of the established country to turn it into 16 autonomous nation states under Habsburg rule, the United States of Greater Austria. And even if the Archduke never could decide which way to favor it was still always clear that he was very much pro-Slavic.

Not to forget the fact that in 1900 he married Sophie, who stemmed from an old Bohemian lineage, out of love and despite the fact that it was morganatic, meaning their children would not be considered in line for succession to the Austrian throne.

So obviously not in any way acceptable in Nazi ideology.

The answer is there in your quote.

The Royal House were becoming Czech in every possible way; and it was the hand of the goddess of eternal justice and inexorable retribution that caused the most deadly enemy of Germanism in Austria, the Archduke Francis Ferdinand, to fall by the very bullets which he had himself helped to mould. And he was the chief patron of the movement, working form above to make Austria a Slav State!

The Habsburgs were Archduke of Austria, King of Bohemia (Czechs) and King of Hungary. Czechs are Slavs.