The thing is that paying for BYOD is currently taxable income. So I you pay them $50 BYOD that needs to go on their income statement.
A company that gives employees a "budget" with which to buy their ideal device or a device from a "whitelist", doesn't have to do that unless the employee leaves with the device before the device has zero value. Now the question arises as to when the device has zero value and that's a tax accounting question that you need specific advice on. One way I suspect its done is that the company uses the 3 year accel. Depreciation allowance. (which is why a lot of companies are on a 4 year hardware refresh cycle).
OTOH, if you don't do a BYOD stipend, I have no incentive to bring my device in. And I will in fact expect you to provide me with a device appropriate for doing my job, just like you would provide me with a desk.