Peter raises a good point. The behavior of the crowd may be more of an issue than the skill of the musicians at the Wiener Mozart Orchester performance at the Musikverein. As wmt1 points out, the musicians for the group are accomplished. The Großer Saal is a beautiful place to experience a performance, too -- it's considered one of the top concert halls in the world. Edit: Having said that, I think it would be hard for me to get past the clothing to enjoy the music.
The Wiener Konzerthaus also has performances aimed at a non-tourist crowd. If you arrive a night early (6/16), you could catch the Staatskapelle Dresden (https://www.konzerthaus.at/konzert/eventid/57260). On 6/19, you can catch the ORF Radio-Symphonieorchester Wien (https://www.konzerthaus.at/konzert/eventid/57265), but it is performing distinctly non-classic-Vienna works.
I've seen positive comments with regard to the Karlskirche concerts, too.
The challenge in Salzburg is that, in June, the city is gearing up for the Salzburg Festival that starts 18 July and runs through the end of August. So, there is not a lot going on in the non-tourist music venues in June. The touristy venues are not bad, though. Like organizer8, I have been to one of the Hohensalzburg chamber music concerts and found it satisfying (full disclosure: I'm not a music teacher and have no formal training, but I have season tickets to my local philharmonic and prioritize attending classical performances when traveling). A number of people have also commented here that they enjoyed the Mirabell Palace chamber concerts.