Looking into your radiator will tell alot about its condition, if you
see corosion on the fins a chemical flush is best, corosion means the
coolant was neglected and is closing the radiators cooling ability. Im
probably wrong about not running the motor when flushing with a hose.

Your method won't clean the radiator because water will come out of the
drain plug at the bottom when it passes through the engine and returns
to the radiator.

Make sure your brand of distilled water doesn't have "minierals added
for taste." Or it'll defeat the whole purpose of using steam distilled
water.

Letting water run out of the upper radiator opening will solve the
problem, but you'll have water all over the place without another hose
to guide it off safely. If any gets into your alternator or other
electrical parts without drying up completely first, then those parts
may be fried (BTW, you do know most alternators carry live battery
voltage even when off, right?)

Or with the thermostat out (everything else in place including the
gasket to prevent leaks) you can fill up with distilled water and run
the engine for a minute (before it warms up, so you don't add cold
water to a hot aluminum engine). Repeat a couple more times. The water
should be pretty clean by the 3rd or 4th time.

It's a lot of extra work if you have a pristine cooling system. I
"drain and refill" every year to year-and-a-half so everything looks
nice and clean. Be sure to account for the extra distilled water left
in the system and adjust the amount of concentrated coolant
accordingly. Use total system capacity as a guide in calculating how
much concentrated coolant to pour in.

Note: if you can only get pre-dilluted coolant (like the toyota pink,
not toyota red) in your area, then the recommended procedure is to use
the pre-dilluted coolant to "flush" your system as if you're using
distilled water!! (It must be an accountant who came up with that
idea). Otherwise you would be left with less than 50% concentration,
which is not good at all for the aluminum. The pink organic acid
coolant is pre-dilluted with 50% distilled water so the grease monkeys
at the dealer can't get it wrong or trying to get by on the cheap with
tap water, because organic acid coolants don't really like tap water.
So look before you add water to the coolant, it may be pre-dilluted.

Check where the cooling tubes are welded to the header plates when you
look into the filler neck. Make sure the solder isn't corroded and
begins to "bloom". Modine Radiator used to have nice pictures on their
web site to tell you about what to look for, but now they merged with
Transpro to form Proliance and the pictures are gone.