And go next door.
This strategy has served me quite well for touring on and off the bike.
Obviously there are iconic places with no equivalent next door, but most have a smaller, less showy, friendlier and more accessible little brother nearby.

without putting it or thinking of it in that way, this has certainly been a big part of my enjoyment of bike touring over the years, exploring and going through smaller places, in a way specifically avoiding the "popular" places because of all the downsides.

As you say, there are some places that are iconic and neat to see (I biked past Niagara Falls last summer, glad I did, despite the crowds) but smaller places and better interaction with people that are fed up to here with "the masses" are what I find more interesting, especially for the one to one interaction with folks.

Or go off season. I tour during the "shoulder seasons".

+1

That's when we like to travel.

England is Small . OP can get places quickly on the Rails , even the Chunnel is convenient.

Lots of Europe's Companies all schedule the month of August to send everyone on their (paid) holiday time.