No.
Indeed OeBB is a major transporter in Austria-and they also run many buses (a few 1000)-this as result of former merger of POSTbuses (in the distant past used to transport both the people and the mails) and BAHNbuses. ALL these buses are now organised by the ´Länder´ (think of as provinces/regions) and most have a livery of that region. The regions set the fares-many have quite attractive day- or other tickets. Also in most tipical holiday regions the ´Guestcard´ you get when staying may vover free use of local buses. Similar situations also exist in Germany, and in the Swiss the yellow POSTBuses are very famous.
In Salzburgerland on FRI all transit is free for now-untill finished- as ´petrol free´ days. This includes several Postbuses lines to famous places like St. Johann or St. Anton-am Wolfgangsee.
In Steiermark and Kärnten are very cheap sat/sun daytickets for 11€
In Tirol there is a 7-day weekly for all transit (incl the trains IN the Land) for about 44€.