I am pretty sure the Presidents Cup is a PGA Tour run event. While I disagree I do understand why players who are members of a competitive tour are not on Presidents Cup teams.
In the US the Ryder Cup team has no affiliation to the PGA Tour. That is why Brooks was on the team. In Europe the Ryder Cup team is run by the DP Tour. Some of the players resigned their Euro Tour membership or are under suspension (I think for not paying fines). I believe there are some LIV Players who still have good standing with the Euro Tour. That would enable those players to be on the Ryder Cup team. I have no idea what Rahm’s status is.
In the US the Ryder Cup team has no affiliation to the PGA Tour. That is why Brooks was on the team. In Europe the Ryder Cup team is run by the DP Tour. Some of the players resigned their Euro Tour membership or are under suspension (I think for not paying fines). I believe there are some LIV Players who still have good standing with the Euro Tour. That would enable those players to be on the Ryder Cup team. I have no idea what Rahm’s status is.