There is another problem with using your own phone for work: many companies now require you to install their MDM profile to access email or other services. This gives them a lot of power to do whatever they like with the device, remotely. If a security breach occurs and they deem it prudent, they can wipe the phone remotely, for example. And I don't even know what they can access from the phone's content, probably everything.
It's quite simply an issue of ownership: if they don't pay for the phone, they don't get that. If you want root on a device, pay for it and issue it to the user.
So, in that sense a device issued by the employer is better. I'm not installing anyone's MDM profile on my phone, that I pay for, that contains my personal photos, messages, etc.