Why can't we detect infinite loops in the busy beaver problem?
The state of a TM includes its tape contents. Since the tape is infinite, the number of different tape contents we may encounter is infinite as well. So a TM is not required to repeat a state in order to never halt.

As a very simple example, consider the TM that will in each step write a character on the tape and move to the right. -- It will never halt but also never repeat a state.