We are looking for a quiet and beautiful village near Paris (1 hour on train) to stay in for 2 or 3 nights in August. We would love a house (3 adults) with a pool (in this beautiful village), that we can relax around. How would I go about finding such a place?
You might start with Gites de France
Moret-sur-Loing. It’s on an RER line. How to find the place with pool maybe do search with the name of whatever village you decide on along with “vacation rental” and “pool.” I am also linking to an article on beautiful villages near Paris which is how I found Moret-sur-Loing last year. https://parissecret.com/en/top-picturesque-villages-to-discover-just-a-few-steps-from-paris/
Can't help with the pool or swimming details , but here is a bucolic place I've known about for a long time , having been a musician as my career . I am thinking of a day trip there on my upcoming trip to Paris this coming Autumn . The English composer Frederick Delius lived there - https://en.wikipedia.org/wiki/Grez-sur-Loing