If they're searching your phone, it's likely that you're already in secondary inspection. At that point if you're an alien they're already suspecting that you're inadmissible and they are trying to build a case. They will ask a lot of questions in secondary inspection. If they search your phone and find that it's wiped, they will ask about it and they will expect a convincing explanation.
CBP officers working in secondary inspection to this all day, every day, they know what they're looking for, they will lie to you if necessary and chances are that they have heard the same explanations many times already. Secondary inspection is not a pleasant experience at all and you don't want to make it worse by trying to outsmart people who do this for a living.
This is definitely very scary and I didn't mean to say that I would try to outsmart them because as you say, they do this for a living and I'm not a professional liar/spy.
However I think my case still applies in the sense that I would expect a lot of people to also buy a burner phone for a vacation trip just as a way to avoid loss in the event of a robbery, instead of as a way to try to outsmart border agents, so I would hope I would not be alone in giving them this reason as to why my phone is clean.
You raise a good argument though. If I'm already in secondary inspection I guess anything you give them or fail to give them will in any case be used against me, basically depending on the mood of the agent at that time.
I guess an interesting statistic to know (not that government would publish it willingly...) would be how many people that went through secondary inspection where denied access and how many of them were not.
Out of the 375 million passengers processed in FY 2014, 34 million was referred to secondary inspection and about 223,000 was found inadmissible. Keep in mind that these numbers also include US citizens who cannot be found inadmissible, but could be referred to secondary inspection for other reasons.
So ~10% get secondary inspection, and of those ~0.65% where found inadmissible (non-citizen as you point out).
Not sure how to feel about it. On the one hand 10% sounds like "not so bad" in that it's actually a minority getting secondary inspections.
On the other hand... if I'm waiting for border agents to let me through and 1 in 10 will get secondary inspections, which by the looks of it would include access to your accounts and such, it sounds like a terribly high percentage.
CBP officers working in secondary inspection to this all day, every day, they know what they're looking for, they will lie to you if necessary and chances are that they have heard the same explanations many times already. Secondary inspection is not a pleasant experience at all and you don't want to make it worse by trying to outsmart people who do this for a living.