It’s kind of surprising what is allowed and not allowed. One problem is that the iOS Instagram client makes it essentially impossible to navigate through a big photo library, you can search and it is really intended just to load the most recent photos. Plus it’s a tiny screen to scroll through 7,000 photos and once you click, you lose your place and have to scroll and scroll.
Sigh, apparently, it is quite hard to make this work correctly. The instagram.com web client doesn’t allow uploads. There are apps like Flume which does this, but they have been kicked off the App Store, so you need to download with home-brew or from their websites. I’m guessing this is primarily to protect ad revenue, but inconvenient.
The best solution seems to be to use Windowed.io which provides a mobile browser framework that runs on the Mac. Instagram then runs in the window and you can save the images, etc.