Adobe have announced the end of support for the Adobe Flash player after 12/31/2020 and will block Flash content from running in Flash Player beginning 1/12/2021.
Since the Brother Control Center 3 uses a Modern style skin that requires Adobe Flash, it will no longer function after this update.
Most likely, this is what the issue looks like now when you try to open the Brother Control Center 3.
You do not need to uninstall the Brother Software Suite, instead, follow these steps:
Here is a video guide for the above method