Ain't no symbol like a children's toy symbol, because a children's toy symbol is… reflective of the way the protagonist and/or main characters remember their youth and continue to be affected by it.

Okay, that's not the pithiest phrase in the world. But it is 100% truth.

It becomes clear midway through the movie that Bertie's stutter is somehow connected to the stress he experienced in early childhood, and the way his domineering father told him what to do. For example, Bertie is a huge fan of models—even in adulthood. But when asked if he played with them growing up, he answers,

BERTIE: No. Certainly not. Always wanted to build models. Father wouldn't allow it. He collected stamps. I had to collect stamps.

Oof. That kind of breaks every rule in the Parenting 101 handbook, doesn't it?

Building models is a sign of everything Bertie had to miss out on because of his strict upbringing. As a middle aged man, he still yearns to do the things he missed growing up, and his stutter seems to hang around whenever he brings up his childhood deprivation. Lionel Logue even manages to use a model airplane to bribe Bertie into singing a tune for him, saying,

LOGUE: Well, as a little reward, you get to put some glue on these struts.

The fact that Bertie agrees just shows how much of a hold those early childhood memories still have on him. And the fact that Lionel Logue is conscious of Bertie's childhood deprivations and uses childhood activities as an incentive for Bertie to keep talking showcases just how much of an excellent, nurturing friend Lionel is.