I thought I heard that, without some of the provisions in the Patriot Act, intelligence agencies would not be allowed to share information with other agencies. But the President is the head of the executive branch, so why can't he sign an executive order allowing the agencies to share that information?
They need the Patriot act because otherwise, legislation passed in the wake of Church Committee hearings in the 1970s completely hamstrung (where not outright prohibiting) any such sharing.
Please note that the Church effects weren't only in explicit prohibition in terms of direct letter of the law - for example, there were chilling effects on information sharing due to heavy oversight and consequent leaks by the overseers, which endangered sources and methods.
Prior to the Patriot act the CIA and the US Military were not allowed to share intelligence with the FBI or any other US Agencies. The reason being that the CIA and Military were concerned with intelligence and criminal activity outside of the US while the FBI et al., was responsible for activity inside the US. Some of the limitations that protect citizens from illegal search and seizure by US Agencies are not limitations of the CIA and/or the US Military. For that reason to ensure that cases were not tainted, and to help prevent the disclosure of US Intelligence assets and techniques, the agencies were not allowed to share information. This developed into a culture where all agencies tried to limit or refuse to share information even those that were allowed to like the ATF, FBI, and DEA.