Ann confronts Kate whenever she tries to pretend her son is alive. This creates tension for Kate and the whole family.
Due to all this, Joe is not too enthused when Chris says he wishes to marry Ann. Since Chris is Joe's only son he knows he can't lose him. So Joe, reluctantly, agrees to keeping the marriage a secret.
Joe keeps Ann and Chris' relationship secret because while he think there is nothing wrong with it, he knows it will hurt Kate. Joe lying about who decided to sell the faulty parts might affect Chris and Ann's relationship.
This mix of traits leads to Joe being dishonest and pretending that his son is still alive just so Kate can keep believing this
Due to Ann's ability to face hard truths like the death of Larry and Chris' ability to be honest and accept this fact as well. These two characters get along very well.

All My Sons

Joe Keller and Kate Keller

Joe is the typical 'American Dream' husband.
Who cares so much about his family.

He cares so much about providing for his
family. That in order to do this he is willing
to be dishonest

Kate is seen as a simple loving mother

But she cares so much for her son
and that makes it hard for her to
believe he is gone.

Chis Keller and Ann Deever

Chris is more honest then his
father. Miller calls him "capable
of immense affection and loyalty"

He loves Ann

Ann is very much the opposite
of Kate. She believes that Larry
is dead and has moved on.

She loves Chris

Joe Keller and Ann Deevar

Without his wife, Joe is willing to
admit that Larry is dead.

Since Joe cares so much about
keeping his family together. He lied
about who decided to sell faulty parts.
He blamed Ann's father.

With or without Kate, Ann has moved
on.

Ann is very no-nonsense when it
comes to dealing with hard truths.
She cut off all communication with
her father when she was told he
sold faulty parts.

Kate Keller and Ann Deevar

Kate is afraid to believe her son
is dead.

Ann has moved on and is not afraid
to say so.

Joe Keller and Chris Keller

Joe wishes to keep his family together
and even though he does believe Larry
is dead, he knows Kate does not.
Joe is also very big on legacy and wishes
to pass his business to Chris.

Chris like Ann has moved one and
thinks his mother's insistence that they
believe Larry is not dead is ridiculous.