Alrighty so once again don't know if this is debunked or whatever but I have an idea as to what two handing the axe does. Link

Whenever I answer the phone I've started to immediately hug my back to the wall while two handing the axe. Because of this the Family Man never shows up. I can make it through pretty much the entire house like this.

I included a video of me messing around with things (sometimes I might stop to write things down sorry) as well as this idea towards the end. The reason why I'm posting the video uncut is because I might have missed minor things.

Around the time where I make my way to the secret passage I stopped hugging the wall and holding the axe. I got around the house like that for maybe a minute or two before captain yellow jacket showed up and welcomed me to the family.

I didn't look to far into it but the hugging your back to the wall thing might just be a glitch and the game eventually catches up to correct the problem when you stop doing it, but I don't know.

The point where I pick up the phone is at about 8 minutes 40 seconds in. Prior to that I had an idea that you're supposed to pick up the phone when your flashlight flickers or something hence the reason I waited so long to pick it up. Did nothing in the end but eh.

On a side note I've also noticed that the controller will randomly vibrate at times. Maybe to signal that:

-You're close to something important. -Family Man is close by. -You're supposed to use an item there. Or -You've triggered an event to happen.

Other than that I don't have much to provide. Sorry I'm bad at investigations but that was my interpretation as to how two handing the axe came into play. Might not even be connected given I hugged the wall the entire time.