I rarely use Chrome, but from what I remember it doesn't natively open PDF files in the browser. Perhaps at one time it might have had a pdf engine in the app, but It might be it got removed. Perhaps that's why they went with the Google PDF viewer.
With Dolphin on my tablet, it doesn't directly open up PDFs either, but rather throws up a notification that the download has completed and I when I click on the notification, it asks what app I want to open the downloaded file with (I haven't set a default). If I choose Foxit, the pdf file opens up, and when I click the back button, I get returned to the Dophin browser. I'm not sure if this is also what you see when using acrobat.
Sorry, I'm not going to be much help to you because like I said, I don't use Chrome. I'll do some reading but not sure what'll turn up.