Rich, assuming you have actually driven the car a distance to get everything operating as they should and that the thermostat is opening and you have no air locks and checked the heater was working fully, car runs at usual temperature on reasonable length drive.
Assuming all that best way to clean the cooling/heating system is to clean the heater matrix, engine block and radiator inside and out, same for any hoses you can. If possible it can be better to remove the rad and heater matrix if reasonably possible (and any pipes and hoses).
The exact details I don't know on your car but small drain holes are nor best to drain from when things need cleaning disconnecting bigger hoses with get more debris out. Existing drain taps or drain holes out to clear them may also be necessary.
The following is for other British "classics" but you can get the idea and adapt as required. -
More thorough system clean and overall -
before starting consider if the water pump and heater fan motor need replacing and (if not already fitted) whether to install an electric radiator cooling fan as this would dovetail in with the work below.
You may want to check or change the hoses, thermostat, heater tap, fan belt and pressure cap.
Procedure:
a) clean the whole system with something like Holts Speedflush
b) drain the whole system - engine block, heater matrix and radiator
c) use a piece of thick wire to clear out the crud that collects at the engine block drain hole at each draining of old coolant, cleaner and when flushing out
d) get the radiator and heater matrix out of the car to give them a good shake at the same time as flushing and reverse flushing [or do three points below with items connected together, flush, back-flush, flush again as below]
flush clean water through each of the three areas until water runs out thoroughly clean
reverse flush each of the three areas until water runs out thoroughly clean
final flush through each of the three areas until water runs out thoroughly clean
e) gently brush clean both sides of the radiator fins
f) gently brush clean the heater matrix and if required renew the foam seal around the heater matrix and the heater box foam seal
g) siphon out the coolant and clean the inside of the expansion tank – remember after to half fill it with coolant and to replace the pressure cap before refilling the rest of the system
h) follow the refill instructions from the Driver’s Handbook to avoid getting air locks or ‘hotspots’ that could cause overheating of the engine.
Flushing and reverse-flushing -
I’ve found just using an ordinary ½” (15mm) open ended plastic garden hose is ideal for fitting to the heater matrix inlet and outlet and an off cut of the same hose is ideal to use as an outlet for coolant and flush water going into a bucket – that way you can see and inspect the crud and muck that comes out and also capture the waste liquid.
You can easily and quickly swap over the position of the hoses on the matrix inlet and outlet for reverse flushing and you don’t need to secure the hoses with clips unless you have particularly high-pressure cold-water mains, if so also don’t turn the tap on too far, you want to clean not damage.
If you are particularly worried about electrics getting wet then cover them in cling film (I only bother to do this if I’m cleaning the engine bay and will be fully rinsing the engine bay).
For the radiator you might want to use some sort of adaptors or just rags around the garden hoses if the radiator is remaining in the car.
For the engine, again depending on where you use as access and drainage points, you might want to use some sort of adaptors or just rags around the garden hoses as hose seals.
On each drain and each flush I like to syphon or blow out the residue liquid from the matrix, engine, radiator and bottom metal cross pipe as a very surprising amount of liquid is left in despite your best efforts to drain – I had the radiator out of the car and shook it every way yet there was still some liquid left in it.
I blow the water out by just using an off-cut of garden hose and my lungs but don’t overdo it as you could hyperventilate.
For syphoning I used a very simple plastic syphon bought off eBay (you can get then for around £4 onwards) which I’ve used for various jobs on the car over the past few years so well worth the investment.
After a thorough clean like this if you use the correct coolant mixture and regularly change this coolant when required (usually every 2 years) then future changes should be just drain, flush and refill.
Some different products (no order) -
Wynn's Radiator Flush - https://www.wynns.uk.com/product/radiator-flush/
STP Radiator Flush - https://www.stp.com/uk/product/radiator-flush/
Liqui-Moly Radiator Cleaner - https://www.liqui-moly.com/en/gb/radiator-cleaner-p000197.html#1804
Prestone Max Total Cleaning System Cleaner - https://prestone.com/product/prestone-max-total-cooling-system-cleaner/
Prestone Toatal Cleaning System Cleaner - https://prestone.com/product/prestone-total-cooling-system-cleaner/
Holts Speedflush - https://www.holtsauto.com/holts/products/speedflush/
OEMTOOLS 87009 No Spill Coolant Filling Funnel Kit - https://www.youtube.com/watch?v=6IVaxY1yXI8
HTH.