I agree that life isn’t fundamentally different from all other matter, but the reason we have the word at all is because it’s handy to be able to encapsulate it as a reference class.
I think the reason we have the word is more to do with historical vitalism than with carving reality at its joints. These days it’s most often, or at least most rigorously, used to indicate “that thing biologists study, with all the membranes and nucleotides and amino acids.” If you want to be more abstract than that, I don’t think trying to define “alive” is really a good approach at all, because it means too many different things. But you can certainly look at all the interesting things it can mean and figure out which one you’re currently talking about. You might want to give it a new name, though.
I think the reason we have the word is more to do with historical vitalism than with carving reality at its joints. These days it’s most often, or at least most rigorously, used to indicate “that thing biologists study, with all the membranes and nucleotides and amino acids.” If you want to be more abstract than that, I don’t think trying to define “alive” is really a good approach at all, because it means too many different things. But you can certainly look at all the interesting things it can mean and figure out which one you’re currently talking about. You might want to give it a new name, though.