I think we all can agree that the current setup does not discourage camping at the health and armor pickups because it's profitable just to camp at the most convenient pickups (like the ones at Union Station or whatever it's called in LS). Most people don't want to forgo armor, nor do they want to go to an alternate location, so they keep getting killed at the convenient spawn. It's so bad that as a terrorist, I can often ignore armor entirely and kill the President without it, since the police are all tied up killing people at the armor spawn who think they need it.
I think disabling the last-used pickup on a per-player basis until another one is used would help. For example, a Union Station pickup would not respawn for me after I used it until I used another one. In order to get the Union Station health pickup to respawn for me, I would have to use the one at the hospital or elsewhere first. Meanwhile, someone who used the hospital pickup last would be able to use the Union Station one but not the hospital one again (since it's on a per-player basis). Similarly, in order to get the Union Station armor pickup to respawn, I would have to use a different one first. I guess this would reset on dying, but that might make it forgiving enough that people continue camping at the station anyway, so maybe not. If people still camp at the convenient places, make it so the system doesn't reset between spawns. This would force people to go to different pickups when they spawn or forgo armor, which would deprive the campers of their flow of targets.
Since the whole point would be to discourage players from fighting at the pickups instead of playing the game and going after the President, the President's pickup usage wouldn't be directly affected by the system. In other words, the pickups would remain as they currently are for him-- he can't ignore the gamemode by camping at pickups because the entire gamemode revolves around where he is. However, if he does choose to camp at a pickup area (he stays within a certain distance of the pickup for a period of time) or uses the pickup, it gets reset for everyone.