I just did a search (using https://tablet.otzar.org, as I don't currently have access to my academic resources), and I couldn't find an instance where this specific phrase (להחזיק יד, and its variants. My search query was `&??חזיק{0}יד`, without the backticks) is used in the meaning "to strengthen the hand". All I could find was of the "to support" meaning. Can you point to specific references where this isn't the case?