Like you said, Bing has their own search index. Why isn't everyone using Bing?
According to estimates[1], Google indexes ~60 billions pages and Bing is only ~4 billions. If these numbers are close to reality, Bing index is 7% of Google's.
IMHO, this leads to all the problems people complain about Bing. The algorithm is shit because they have less data to fine tune it, people use it less because the algorithm is shit or the results don't show up, so they cannot collect data to fine tune the algorithm more, ...
When Google was just starting to gain popularity its search results weren't much better than Altavista's. Yet, its interface was much cleaner, it loaded and spewed out results faster, the overall experience was better. Bing is the Altavista. The user is bombarded with cookies, logotypes, ads, maps - all that complex layout right from the start. As a user you feel napalmed by information, and nobody likes that.
I do remember yahoo's home page was terrible, although I think there was a clean search.yahoo.com page no one knew about. Back then (and even now) having the search page load fast was pretty important so this always seemed like a nasty own goal.