Now that CPU process data byte by byte, what's the reason to use bitwise & bitset?

Of the reason of manipulating bits, the only one I can think is about saving storage. But now that CPU don't even have address for each bit, AFAIK CPU have address for every byte, then how can bits operation saving anything? If it cannot, then why we use it? Thanks.
You need bitwise operations because that's the only way to access bits. They acn't be addressed directly because the machines are byte addressable.
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