I use both. Yes, AM swears by it, but then, he doesn't use Lightroom. He downloads with Downloader Pro. He has gone back and forth over the years between using ACR in Photoshop and developing Raws in BB. I don't know what he currently uses, but I suspect it's not BB as he hasn't really mentioned using it in his blog or newsletters.
That used to be my workflow, too. I still have Breeze Browser, and I use it for limited purposes. All my importing, selections, Raw developing and output is done in Lightroom. Needless to say, Lightroom is my "DAM" (digital asset manager).
ACR, as found in Lightroom, is far more capable and robust than Breezebrowser. I won't go into all the advantages of Lightroom; I'll just say that I mostly now only use BB for a quick check of jpegs that I've exported, just to make sure that what I'm about to send out is what I think it is. Belt and suspenders.
In addition, not every picture on my system is imported into Lightroom....other people's shots, random pics from email or the web, these kinds of things I view in Breezebrowser. It's quick, and since it is a direct file system browser, it shows me exactly where the file I am looking at/for is located.s
I didn't upgrade to the new version, the one requiring yearly subscriptions....ver1.9.3 and still going strong!