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There are also plenty of occasions where a memory byte is set to $FF as a flag value; with an "M" ("A"? "O"?) flag simply loading the byte into a register would be enough to tell.
or BIT, so you don't even have to affect A, X, or Y. There might be a use for an F flag, but it's common to use a byte as a flag and anything but 0 means the flag is set, or at least anything negative. Without an STF (store FF) instruction which I would like to have, setting a flag can be a simple DEC, as long as you don't do it so many times that it again becomes 0 or even just positive. Use STZ to clear it. Then BIT, BMI/BPL for example tests it, without regard for what's in any of the registers, and without affecting them.
INC and DEC don't affect the carry flag.