Does the Hot Tub in the Castle allow me to improve relationships?
Solution 1:
The only real use for the Hot Spring is upgrading it to max to get an extra character - an Onmyoji called Izana on Birthright and Conquest, or a Master of Arms called Fuga in Revelation. They join at different points in each route.
Other than that it has no use other than as a "special interactions" space, most notably if the Player Avatar has a spouse and/or children. Interactions there have no effect on relationships.