A rare outage should cause an executive to think "let's make our own search engine index"? If for every partner outage i would go "let's do it ourselves' i would be long out of business
If DDG relies mostly on Bing and it fails forever in the future, they can "simply" make a contract with google and continue whatever they were doing.
Same for OpenAI. Its not that Bing is the only search engine index on the web is it ? So yes, it would mean they have to spend a lot of manpower in a short time.
On the other hand, how likely is it that Bing just goes offline?
Do you always have a second datacenter in case your provider fails? Not everyone does it and as long as you have a way of putting things back up in reasonable time you are good to go.