Wondering if starting our tour late August would have us adversely affected by local sights and eateries being closed or being really busy with European tourists. Looking at August or September.
Where are you planning to go? Generally September is better. Either way eateries and local sights in Austria will not be closed. The only venues that will not be around in August (in Austria) would be orchestras and opera, the exception being the festival in Salzburg of course..
I wouldn't worry about facilities being closed in Austria in August. Kids will be heading back to school so the crowds will drop at the end of August and into September. I would think that you will have a wonderful time on either week.
For Austria, my preference would definitely be Sept. over Aug. Better weather, less crowds, wine harvest season, lots of fun festivals, better music options, etc...