In my experience (I worked with BERT and RoBERTa), not updating the transformer model parameters during fine-tuning resulted in lower accuracy and slower decrease in the loss value. This might mean that the fully connected layer alone is not enough to model the task at hand. I suggest updating the parameters of the DistilBert model as well, which is what fine-tuning is for.
I should also note that freezing the first 6 layers of BERT-base did not decrease the accuracy of the model significantly, in my case.