OK, I ventured a down that rabbit hole somewhat, and it turns out you are, in a sense, both right.
In fact, phones had, and still have, a real time clock (RTC) that is able to keep time even with the device powered-off.
However, as I learned from here and here, the RTC on Qualcomm devices is read-only. Upon the very first power-on, it initializes itself at 1st January 1970, 00:00:00 UTC ("Unix epoch"), and then just counts from there onwards through the lifetime of the device.
Indeed on my (rooted) Pro1, the RTC clock returns
rostkatze1:~ # hwclock -r
1973-09-11 08:31:30.407