Your version suggests two actions happening together. Or at least overlapping. (At least to me. But some will say your sentence is ungrammatical if there is no overlap in these two actions.)
Meanwhile, the original suggested two sequential actions.
Your version strikes me as very standard, preferred by many authors perhaps because it saves a word or two. And most authors wouldn't care about that small difference in meaning. But that brings us back to the same topic we have been dipping in for months, choosing verbs for style versus slightly better meanings.
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