Shaolin was a sect, different from the Shaolin Temple. Perhaps they were similar a long time ago, but now there were too many guests in the Shaolin Temple, so it was not suitable for practicing martial arts, so they were separated.
There were also traditional monks in the Shaolin Temple, but there were very few of them, and their Kung Fu was far less powerful.
For example, the man they had just met on the road, many monks might not even be able to beat him.
In the end, Li Tian didn't give Vermilion Bird a reason for her request. He decided to treat her injuries first and wait for her to recover.
The Vermilion Bird really wanted to become stronger. It was already good enough that she could reach Li Tian's level.