|Uh-oh, the refugees are coming!|
So in summary, there is no evidence the medieval mind considered or knew water to be unhealthy. Bad water was a concern, but people had their own guidelines on how to tell good from bad. If you want more evidence, look at how long it took Dr John Snow to persuade people that cholera was spread via water. That was in the 1800s -- more than two centuries after the Mayflower -- and it was also when they realised that drinking beer was safer than well-water.
Plus, boiling the wort for beer before fermentation seems to have come in some time after the introduction of hops (remembering here that in mediaeval times, ale was unhopped, beer was hopped). Boiling modifies the acids in the hops and is needed to fully activate their bitterness and preservative qualities. Before hops, there was no need to boil.
What you did need was hot water for the mash, as it's how you get the fermentable sugars out of the malted grain. This heating would have been enough to kill most bugs, but was not a boil. Apart from anything else, at a time when a fire meant collecting or buying firewood, unnecessary boiling would have been a waste of expensive and/or scarce fuel.
The truth seems to be that drinking water was not fashionable among the middle classes in 1620 -- it was something that only poor people did, and the religious fanatics aboard the Mayflower were not poor people.