Lots of us have done that. And I think you mean dispatching instead of culling - most folks use culling to mean either eating or rehoming, but basically they're not part of your flock any more, but not necessarily dead. Dispatching means you sent their spirits to God, and processing is the whole process of dispatching thru preparation for cooking, and can include parting out the meat or separating the carcass via spatchcocking.@BigBlueHen53 have you ever culled then froze a bird to process at a later date?
Culling and plucking by hand takes a significant amount of time and this time of year it seems unlikely I'd get around to processing before the sun goes down. I tend to put off culling until I have a full day to dedicate to the whole process. I'm sure with experience it'll require less time. If I end up processing the majority of my cockerels from hatches I'll invest in a plucker, but so far I've managed to rehome them.
I process one or two birds at a time - they take me about 1.5-2 hrs each, depending on if I skin or pluck. I put whole carcasses in the fridge and rest them for days until rigor passes. Alternatively you can put them in the freezer and then take them out later to age them before cooking. Either way, I'd recommend you remove feathers and entrails before freezing - I wouldn't freeze something you aren't ready to further process (cut up and/or age) prior to eating. I don't process outside when it's cold enough for my hands to freeze. If you have to do it, I'd do as much as you can outside, have a bowl of really hot water to warm your hands up in occasionally, maybe two with one soapy, and then get inside to finish parting out the carcass as soon as possible on a cutting board or something. It's not so bad when skinning (nice warm carcass if you do it in less than 30 minutes helps keep your hands sorta warm), but the washing of the hands and knives is not pleasant.
Freezing with entrails can make it take longer to get the bird down to 40F, you want to do that part as quickly as possible to reduce bacterial load. Also, don't stack them up at all, give them their own place with lots of coldness so you don't have warm spots or overload your heat removal capacity of your freezer.
In England, it was common practice at one point to hang game birds up for a week in cold temperatures, then harvest the breast meat. You might look into doing something like that (temps are key), but if you go that route, don't go messing with the entrails after a week - I've not found that it's safe from a bacterial perspective. But others might know more than me about that method, as I've never tried it - it's more common when hunting pheasant and such.
Hopefully, you'll get insight from someone who has tried to put whole birds in the freezer, and found it's not a big deal - I've not tried it myself.
