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It will work just fine, with a little understanding of how to set it up.
Suspend the watering line so you can adjust its height as the chickens grow.
Don't put the barrel too high, up in the rafters is too high. Lay the barrel on its side with the bottom side approximately the same height as the nipples. This will give you roughly 0" - 24" water column pressure depending upon how full the barrel is. On its side you'll get a narrower pressure range as the barrel empties as you are not raising the height of the water as high as with an upright drum.
Without an automatic regulator the height of the top of your water supply is the
only thing that sets your pressure. If the water is 24" above the nipples, you have 24" of water column pressure. As the barrel empties, let's say you have 8" of water remaining above the nipples, you now have a water column pressure of 8". Any water stored below the height of the nipples is unusable. When the water level gets to to the same height as the nipples you have 0" of water column pressure and no flow. Water does not flow uphill.
Pipe size has nothing absolutely nothing to do with the pressure at such low flow rates. 3/4" is more than enough to handle the flow. I would suggest a diameter/wall thickness that provides some mechanical strength to it though. You don't want it bowing if the birds decide to use it for a perch.
I'm not sure how to explain that partially closing the supply valve has nothing to do with the pressure. For the most part it is a closed pipe with no flow (except for a droplet at a time). If the valve is open whatsoever the pressure
will equalize across the valve. Only when the flow is greater than what the partially closed valve can provide will you get a pressure drop. That will never happen extracting the water a droplet at a time.