So - a bit of follow-up, not in any particular order.
Regarding LL libraries, they are not "obscure" and have proven to work well while being lightweight (compared to the very bloated HAL). Directly using registers is always possible, but takes a lot more time and the LL layer helps making porting to other STM32 families much easier while also being much easier to read than directly accessing registers. So, that part is irrelevant. Of course, like all libraries, they can have their own quirks, such as these macros not being symmetric between LL_RCC_GetSysClkSource() and LL_RCC_SetSysClkSource(). All libraries have quirks.
Just to give a bit more context, I did not have direct access to the "failing" boards, so that made it more difficult to debug remotely. As mentioned, I first suspected HSE_RDY not getting set (while the oscillator was looking like it at least oscillated), but it turned out to be something else. Indeed the
while (LL_RCC_GetSysClkSource() != LL_RCC_SYS_CLKSOURCE_HSE)
test used the wrong macro, which should have been LL_RCC_SYS_CLKSOURCE_STATUS_HSE rather than LL_RCC_SYS_CLKSOURCE_HSE. Obvious in hindsight but close enough not to be picked up when you're right in the middle of debugging something. Typos happen.
So that explained why the switch to HSE appeared not to work in the minimal test firmware, and that was misleading. But after fixing that, the root issue (that triggered the write of this minimal test firmware to begin with) was finally found. As I said, not having direct access to the boards made it a bit hard to locate exactly the cause. It turned out to be unrelated to the HSE or PLLs, but to the LSE.
Indeed, in the original firmware, there was the initialization of the LSE, but on the tested boards, we had not equipped them with LSE crystals, the pins were just meant to be used as spare GPIOs. So when enabling (due to a config error) the LSE, the OSC32_IN/OUT pins were just floating. The unexpected here (that was misleading as to the cause) is that in 8 boards out of 10, the LSE init code was getting out of the wait loop (for LSE_RDY) without a problem even without any crystal on the OSC32 pins. I certainly wasn't expecting that. My best guess as of now is that slight differences in MCU silicon and possibly flux residues on the boards,etc, could make it pick up noise on the apparently "good" boards that would make the LSE logic set the LSE_RDY flag, even if just from noise, while on other boards (the "failing" ones), LSE_RDY was never set. While it could be just seen as an oddity, could be interesting for someone to test it and see what happens (attempting to enable the LSE with nothing connected to OSC32_IN/OUT) on STM32H7's. Not that it should be seen as a problem either way, just again an oddity but that made me miss this issue at first.
And while debugging this, as we can read various stories of HSE oscillator designs being marginal and not working on all boards, and even that the STM32H7 was supposedly more finicky with its HSE, I also investigated this part, but it was unrelated in my case. I can even tell you that that it looks more forgiving than many seem to say.
So, case closed. With a bit of this odd LSE_RDY thing, which, while it was misleading during this debug session, is of course of no consequence otherwise.