This answer expands on the other answers, which I don't disagree with, to answer a secondary part of the question.

I understand that planning and/or intending to commit a murder is in itself a crime, but I'm more interested in what physical act he might do that was irrefutably a crime (without the need to delve into his mind).

None.

All relevant crimes have an intent requirement.

Even once he fired the shot, there is an intent requirement that requires delving into his mind. If the case had gone to trial, he could have argued, for example, that he was just cleaning his rifle and accidentally happened to fire shots at the President. And, if a jury believed him that would provide a legal basis to acquit him.

Proof is distinct from when someone has first committed a crime. It isn't terribly uncommon for law enforcement to lack proof that someone committed a crime for decades after it is committed, or even never. A huge share of all crimes are never "cleared" (i.e. "solved or rendered moot").

This answer expands on the other answers, which I don't disagree with, to answer a secondary part of the question.

I understand that planning and/or intending to commit a murder is in itself a crime, but I'm more interested in what physical act he might do that was irrefutably a crime (without the need to delve into his mind).

None.

All relevant crimes have an intent requirement.

Even once he fired the shot, there is an intent requirement that requires delving into his mind. If the case had gone to trial, he could have argued, for example, that he was just cleaning his rifle and accidentally happened to fire shots at the President. And, if a jury believed him, that would provide a legal basis to acquit him.

Proof is distinct from when someone has first committed a crime. It isn't terribly uncommon for law enforcement to lack proof that someone committed a crime for decades after it is committed, or even never. A huge share of all crimes are never "cleared" (i.e. "solved or rendered moot").