No, keeping a mobile phone on airplane mode does saves batteries. But, if I am write, you stated that he won't text in front of you, if you happen to glance he turns away, in-other-words in totality he hides the phone and messages from you.
A relationship does not comprise of keeping secrets and it certainly comprises of Trust.
Do you think you have a reason not to trust him?
People have answers for everything - " You don't trust me" is exactly what he would state .
I would be asking why he chooses to never let you see anything the texts to people, even by accident, ie) you are walking past, why he turns away so you can't see. If he were to say "why don't you trust me?", I'd be saying " Why wouldn't I but why would you continually ensure I can never see anything you text even if I am next to you?".