With or without a bucket, you can get free ice delivered (except for tip) from room service 24x7. Try getting ice from a cabin steward 24x7. The zip bags used by room service are vastly superior to ordinary store bought ziplocks. If you ever have eaten a soggy sandwich that was in a regular ziplock bag in a cooler, you'll know why. I would not depend on regular ziplocks (or freezer or whatever).
Going forward, I think I will continue to order ice from room service. It is an improvement.