It is disappointing to read posts that some of the new 16e phones have the same problem as the 16 Pro of not turning on sometimes without using the side button hack.
I haven't seen (or noticed, at least) any posts that the 16 has the problem.
I know that Lawrence has recommended calling Apple support if you have the issue. The problem there (and with the so-called genius bars) is that not all the Apple support people are aware of the problem. Some people have wasted a lot of time with this approach.
The ones that got help seem to be getting replacement phones, sometimes reconditioned. So far, there doesn't seem to be any fix forthcoming from Apple. Nor have I seen any acknowledgment from Apple that there even is a problem.
I may be the only one who started with the problem, did the side button kludge/workaround on three occasions, and then the problem just disappeared on its own. Mind you, I am not complaining!