Writes should be efficient but also minimally prone to corruption in case of power loss.
Is there a risk of losing a whole page of data if you cut-off power to an SD card during writing?
Yes of course. The term used for SD cards is "blocks" rather than "pages", but it's just a matter of terms. The default block size is 512 on modern SD cards.
A few vendors claim - as some have said above - to have protection against data loss in case of power loss - but I don't know how effective that could be. We could only imagine that for that to work, the card would need to have sufficient capacitance on the power rail to allow completing a block write. I don't know how realistic that is given the size of the cards - in particular for microSD cards. SD cards (not microSD) usually have a whole PCB with bigger passives on it, so that looks already more doable.
Because if there is then updating FAT metadata will risk nuking a whole bunch of files, or at least a big chunk of the file being updated, if it's large enough and not fragmented.
There sure is.
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That being said, FAT is the default filesystem for SD cards and lots of cameras etc use it everyday with some degree of success. I think I have even heard that the initial sectors that typically hold FAT metadata may be made to be more durable than the bulk of the card, but don't quote me on that.
As you said, there's a difference between "some degree of success" and "reliable". Cameras and cell phones are big enough that they can embed all the capacitance that would be needed to successfully complete writes, and decent software does the rest. Now if you physically remove the card when it's writing blocks, that's something else.
FAT has been used with "some degree of success" in a very large number of applications, including computers for several decades, but that's definitely not a robust file system. It can corrupt very easily.
SD cards used as a commodity in consumer products is not a good example of reliability. As we pointed out, it's not even possible, apart from a few exceptions, to know if the card's controller does any kind of wear leveling, whereas it's an information you always get (and anyway, they all do) with SSDs.