For the Doctor, just as you said, only has 1 scene which she tests your strength after you took her medical experiments.
As for the Nurse, however, has 3 scenes that requires you losing a battle and get sent to the Hospital automatically.
There are 2 ways for you to lose a battle:
When you don't have either Athlete / Living Weapon Trait activated when you're trying to defeat Mario's Goons in order to save Sofia ( Sofia Love Route )
When you got defeated during the Spirit Hunting Minigame at the Shrine
As to get all the Nurse's scenes, you have to lose a battle and got sent to the Hospital 3 times to get them, which are:
