I want to check out the thermostat in the '59 292 in my Model A.  I have read that chemical cooling system flushes are not great for older engines.  I have watched videos where they use a garden hose to spray out the engine (with thermostat removed).  I REALLY hate antifreeze and don't want to put my dog- or anyone's- at any kind of risk.  

It isn't running hot- quite the opposite- so I have no reason to believe anything is gunked up.  

Am I just overthinking this?  Should I just drain it, check out/replace the thermostat, and refill it with new 50/50 antifreeze and distilled water?