0) Embed apps on the frontend: The APIs you mention are good, but they are backend APIs or APIs for customizing (say) maps within a widget on your site. I'm talking about opening up the frontend too. A prosaic way of thinking about it is an App Store for each large Yahoo subdomain, starting with search.yahoo.com but ultimately allowing very deep apps that pull from many Yahoo properties. Importantly, app installation buttons should be heavily but tastefully promoted: every relevant URL, map result, or email from an app provider should allow a one-click install with ID and billing optionally from Yahoo.
Well, if you've tried making a Gmail gadget, it involves unnecessary XML wrangling, many separate pages of documentation, and is nowhere near as simple as it could be. Stripe/Square vs. Paypal show how a good API can transform a category.
Moreover Gmail Gadgets are not heavily promoted nor a profit center for Google, so they are an afterthought. As indirect proof of this, when pg had that recent thread on fixing email, had the Gmail API (+ Gadgets) been a strong API for working with email, people would have brought that up as the obvious solution.
As for OpenSocial, that seemed to me to be big companies banding against Facebook, not actually opening up the web. But we can agree to disagree here.
Read that GMail API again - it is designed to do exactly what you are proposing. The section you need to read is https://developers.google.com/google-apps/gmail/#gadgets
Both Yahoo & Google tried this previously with OpenSocial. It bombed badly.