I don't think they wanted to flex muscle over Apple, I think they were trying to build case law for situations like this. Also breaking into a phone with an exploit like this is expensive and if they have an exploit, they might not want to publish that they have it in the future so having the backdoor provides deniability even if it's fundamentally dumb.

/puts on tinfoil hat

There is also the other option which is that trust in American tech companies has been sketchy at best following the NSA leaks and this was a chance for the Obama administration to allow companies to reestablish some legitimacy when it came to security by making the US government look evil but having the corporations 'prove' that they are not backdoored by the NSA. They can still break in the covert way, but it makes it look tech companies are not as compromised as the NSA leaks would suggest.