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Does the whole sector thing still make sense for SSDs? Do they still work by sector or can every byte be addressed directly?

The "native" sector size of an SSD is the erasure block size, typically 64KiB.

I'm fairly sure SSD erase block sizes are more like 2 MB or more. Even 8 MB wouldn't surprise me anymore.

current SSDs are page/block oriented, but that's likely to change. (See intel's relatively recent Optane).