TC can open a file from drive, you're just not understanding what it means to open a file. Using aqFile.Open will open a direct filestream. If you know what you are doing you can try to parse your data out of a raw pdf stream. Also opening a pdf in a browser isn't going to open the adobe application. Browsers can natively read pdfs, typically using pdf.js. So when you open a pdf in a browser expect it to appear as just another Page object, adobe is not going to be opened.
If you want to open a file in an application, you're going to have to launch that application first and open the file through the application. TestComplete isn't going to know the how and what of opening every file type that exists. I suppose they could have a built-in function that opens a file in the default program, the same as if you double click the file in Windows explorer. But that function doesn't exist right now from what I know. You could put in a feature request if you think it would be useful. Sounds useful, but it's not too hard to just setup a TestedApp to launch the application you want and then open the file within that application.