I've thought about doing this a lot. I have a solid and stable 20 Mb connection at home, only have had ~2 hours of downtime this year.
The thing is... I don't want to basically trap myself into AWS or Dropbox. I run a 6Tb array like you, and kinda set up my own cloud, drag and drop, etc.
I don't think I'd ever give it up because:
1) It's cheaper to run myself (long term) - I'd have to consistently pay someone else ever increasing amounts to store data (which seems dumb since storage gets cheaper every year, their service doesn't).
2) Security - I don't want anyone to have access to those files but me. I'm 100% sure Amazon or Google would be able to access those files if ordered too by a government. Hell, they might do it without being ordered, who knows what's happened since Snowden.
3) Ease of use - my setup is based on a local network (as well as an ssh component). Basically, my setup works like Dropbox. The only drawback is no mobile component. If I want to change that I have access to all the parts and can do what I want. With AWS (which I use all day for work) I cant swap out a drive or stream a movie to my TV, phone, or desktop with a click.