"the phone's Bluetooth behavior is correct. The Bluetooth turns off after a restart, so that the phone boots faster and the battery is conserved (in case Bluetooth was left on by accident)."
Interestingly, today I discovered that both of these responses were incorrect, and that there is a setting hidden deep within EMUI 5.1's convoluted menu structure which will fix the Bluetooth issue.
In Settings go to Security & privacy:
Then Location services:
Select from the 3-dots menu Scan settings:
And finally turn Bluetooth scanning off.
I have no idea why this allows Bluetooth to remain on after restarting, and apparently neither do Huawei. Note that this also affects other brands. I found this solution on a Lenovo support forum relating to the Moto G5 Plus.