A single, good flush should be enough. The rest is "on the safe side" and overkill stuff. ;)

I don't recommend using solvents but, as long as it evaporates with no residue and is compatible with the cooler materials, it will expedite the process. Use proper PPE and maintain good ventilation when using solvents.

Edit: If a little old fluid is left in the cooler, it won't do much harm (dilution will help protect you). If you don't get rid of all the solvent, it can harm the fluid (thin it out or react with it). A good bake & purge will make sure you get rid of all the solvent.

good point about getting all the solvent out, ok pour in a cup of mek, cover fitting holes with thumbs, shake like hell, pour out, let sit for 10min, close eyes and blow in one hole to get fumes out of core. Notice i said blow not suck, if you inhale the fumes you may get dizzy for 15min.

Edit: possibly even high, and that WOULD be illegal