Ok .. Decided to give it Grub Customizer a try on a VM box so I didn't screw anything up. I was able to change the default kernel without any issues. Here is how I did it.

In the General settings tab I clicked the pulldown for predefined. It listed all the kernels available. I chose the 3.19.0-58-generic and clicked save ... I rebooted into that kernel. Even though I was worried about it putting the name instead of the numeric value .. it seems to work fine.
So you would have no problems doing the same thing .. just choose the generic one for the kernel you want to use .. not the (upstart) or (recovery mode)
In your case .. the pulldown would probably say either
Ubuntu, with Linux 4.4.0-21-generic or
Advanced options for Ubuntu>Ubuntu, with Linux 4.4.0-21-generic