We do this as well, and I’m surprised more people don’t do the same. In July we froze about a dozen bottles while home in Jersey along with a Pack it cooler. They were still mostly frozen by time we checked in to POP, and kept my DH’s beer cold as well! I just wish the refrigerators still had the freezer compartment because after a couple days the ice was gone.
Taking in a small cooler inside a drawstring bag is the main reason I can never go bagless. The freedom of not having to carry anything or wait in bag check is very enticing. I just hate the thought of paying for water, and the tap water gives us stomachaches!