As I've been asked by several people now, I thought that I'd post this here as a reference for owners who don't have and don't want an auto-bleeder on either the radiator or at the waterpump/thermostat housing, but would rather prefer to maintain a stock cooling system and have to mess with bleeding the radiator (or water pump/thermostat housing)
Instead of pulling the hoses off the radiator, I prefer to pull the hoses (9 and 16) off the pipes (5 and 18 ) in the engine bay. If you still have a stock/OEM radiator with the plastic side tanks and flanges/collars like I do, the less you pull the radiator hoses off and on, the less problems you'll have later on down the road with the radiator - at least that's my take on it. I then proceed to drain and back-flush the radiator (by using a spray nozzle on a garden hose on the opening to pipe 5, and using the nozzle with a rag wrapped around it to build up some moderate water pressure) and heater core / engine (by pushing water through the disconnected top hose of the header bottle) separately. Up to you if you want to pull the block drain plugs - sometimes I do, sometimes I don't.
If you're trying to collect the old coolant, as opposed to simply discharging it onto the ground, then you'd run compressed air through the system during the catch phase, followed by your back-flush with water. Be careful not to put too much pressure through the system, but a moderate amount is necessary to do a good job and won't hurt anything.
After the back-flush, I like to blow the system out again with compressed air, using the air to dry the water from the back-flush, as my garden hose water is quite hard and I want as little of it left in the system as possible before refilling. I refill the system only after it's completely devoid of liquid/fluid, or at least as empty as is reasonably possible.
I then put the drain plugs back in the block and reattach the hoses to their pipes, leaving only the top hose of the header bottle still disconnected. Note that I don't remove any of the hoses near the thermostat / water pump.
I then prepare my coolant mixture in a clean 5 gallon plastic bucket. 40% distilled water, 60% concentrated Prestone DexCool Antifreeze, and a bottle of Prestone Cooling System Treatment (it's a combination corrosion prohibitor and lubricant). I prefer mixing it myself as opposed to buying the premixed 50/50 stuff (no good reason for this - I just do) and always mix up about a half gallon more than I'll actually need. Yeah, I use DexCool (the orange stuff) and have been doing so successfully for the last 12 years, despite what others on the forum have said about it. DO NOT use any kind of sealant tabs or a liquid sealant product unless you want a mess and a plugged up cooling system.
I then use an old (previously flushed) fuel pump and prime it with my coolant mixture in the 5 gallon bucket. Once primed (free of air), I hook the fuel pump up to the still disconnected top hose of the header bottle, using a simple barbed coupler and a couple of hose clamps. I then activate the pump and allow the system to fill until the header bottle is practically full (don't worry about overfilling, as the system will purge whatever it doesn't want).
I then reattach the hose to the top of the header bottle and place whatever mix remains in my 5 gallon bucket into an empty antifreeze jug.
I then jack the rear of the car about 3 feet into the air and position jack stands in the regular locations. Set the HVAC mode switch to the heater or vent positions. I then start the car with the rear end in the air and allow it to get up to temperature (thermostat opening and cooling fans engaging). But always be sure to watch the temp gauge and immediately shut the engine off if you believe you're on the way to overheating. You'll want a bucket under the purge hose from the header bottle unless you're just discharging onto the ground, although the first warm up should result in most of the excess coolant in the header bottle being sucked into the system. After the fans have run for a minute or two, kill the engine. Check and top off the header bottle if needed. Repeat running the engine up to temp, stopping, and checking/topping off the header bottle until you no longer have to top off the header bottle or the header bottle starts dumping a significant amount of coolant out the purge hose and into your catch bucket / on the ground. You should only need to repeat once. Doing this with the rear end in the air purges all the air from the cooling system via the header bottle without having to mess with any of the bleed valves.
I don't run any kind of self-bleeder kit on my car - all 100% stock/OEM (with the exception of the heater control valve - original one developed a leak at the pivot).
Put the rear end back on the ground, top off the header bottle to an over filled level if necessary, and take the car for a good 10-20 minute drive, being mindful of the temperature gauge. At the end of the drive, check the header bottle and top off if needed, leaving it over-filled, but you should be done.
Note that if you don't refill with some type of purged continuous flow pump as I do, then you'll have to bleed at the waterpump/thermostat housing bleed valve/nipple. Also, if you remove the block plugs during the drain and flush, be sure to teflon tape and anti-seize the threads before re-installing.
So, that's what I do and it has worked great for me over the years. My car has always run nice and cool, as Steve can confirm (95F ambient temp in bumper-to-bumper slow moving Woodward Dream Cruise traffic for over 90 minutes and my car never breached about 180F).
As always, best of luck and have fun with your DeLorean!