Private browsers don’t exist. Not here, and not in most of reality. Not outside of maybe Tor and a hardened and isolated browser, and a whole lot of care. Particularly not if somebody else controls and monitors the local device and the upstream network. Which is (or can be) the case here. Pretty much everything on that phone can be tracked and logged, or can be blocked, if the owner is inclined to configure the device for that.
Company-owned and company-managed devices are at the behest of the company.
BYOD is its own mess.
Your own device—without company add-ons—is the one that you (and the vendor) control.
This discussion reminds me of depending on the “company lawyer”, should something go legally sideways involving both you and the company. The company lawyer is the lawyer aligned with the interests of the company, and that’s not necessarily a lawyer aligned with your interests. Similar possibilities with this phone. Sometimes the requirements and expectations and configurations all align with both the company and you and your respective interests, and sometimes not. And with a company-owned device (and absent local employment regulations or such specifically addressing the situation), the company has precedence.