Should weight distribution change more when fine-tuning transformers-based classifier?
Its difficult to judge by just looking at the weight distribution. You will be in a better position to identify the situation if you check the following things also:

1) Check if the loss is decreasing during training. It is very unlikely that a randomly initialised last layer will not need training to succeed in a downstream task. This is because even if the initialise layers are producing meaningful values, the randomly initialised final layer will output garbage values in the beginning. If the loss is decreasing then the model training is fine. Alternatively, check if the final if the final metric is improving e.g., accuracy.

2) Check the gradient received by your custom layers. One of the reason that the weights distribution is not changing could be due to zeroed or negligible gradient. I don't think gradients are zero because your plots show the weights to be changing, but still if If the gradient is zero then check your implementation. Something must be off like not using the actual output of the model while computing the loss for backpropagation. If the grads are very very small, try increasing the learning rate of the optimzer.

Conducting the above 2 analysis will strengthen your understanding of the current situation for the model training.