It is doable, but there are a lot of drawbacks to consider. Using an ice box would be a great way of making it work, but I wouldn't repurpose a conventional refrigerator for the job. Their insulation is woefully inadequate. The same really goes for the chest freezer as well, but at least in that situation the cold air doesn't spill out when you open the door. It would probably be best to build your own ice box. I would use no less than 4" of insulation all around in a chest configuration. It would be advisable to have a table, stand, cabinet or similar to get it off the floor and save your back. This will make it easier to get in and out of quickly which would help things stay cool longer. Perhaps even 2 smaller chests
side by side that you can keep more ice in one for 'more perishable' items and less ice for things that merely need to be kept cool.
If you keep the ice out of the melting
water it will last a lot longer. I have heard of modified picnic coolers that have held ice for over a week, so I am certain a well insulated ice box can be made to work. It will probably work better with one or two responsible adults using it as opposed to guests/children/elderly or other people known to not be completely reliable about things like keeping it closed and well loaded with ice.
Using a generator daily for ice seems like a recipe for disaster. I have heard of smaller ice machines that are supposedly faster and more efficient at making ice than a chest freezer. If you were planning to store perishable things in the freezer and not just ice then it would require Herculean willpower to avoid a disaster. The cost of constantly fuelling plus one ruined freezer load of food would probably come close to the cost of going DIY
solar for your refrigeration. If you get sick, injured, have some sort of crisis, or simply want to stay out for a night or go on vacation then it would require multiple contingency plans to keep things operational. Perhaps if you were using some sort of battery setup with an automatic system to start the generator then maybe it would be different, but that seems about as complex and expensive as going solar.
As I mentioned before, chest freezers also have pretty bad insulation. You may be able to find some videos on making a chest freezer more efficient. At the very least you can add some styrofoam to the bottom of the freezer and add insulation to the lid/door on top. If you can locate your refrigeration in a building that is kept as cool as is reasonable then that would help. Perhaps an earthen structure or thick walls with adobe, earthbags, or some other thick thermal mass shaded from the sun would be ideal. This would reduce the temperature difference between the inside and outside of your freezer and icebox which means less
energy to keep things cool. I wouldn't just walk into any house and try to implement this system in their existing kitchen because chances are it would be more of a hassle than it is worth.