People did find out.
Imagine if Snowden decided to just do his work and move on? How much longer would it have taken for these facts to be revealed to the public?
So literally no downside to putting a backdoor and lying about it
Just like lots of people want universal healthcare, a clean environment, an arms embargo on Israel, affordable housing and education, etc.
It can hard to believe these are majority views sometimes, but that's what you get when the entire media landscape is owned by like 10 people.
People did find out.