That's not a good excuse to me. One, Pike could have remained at Starfleet Headquarters, until the Enterprise was repaired. And two, Starfleet could have ordered the Discovery crew, with Saru in command, to continue investigating the red spots. After all, it's officially supposed to be a science vessel. Right? I don't see the need for Pike in command of that ship.
Discovery doesn't have a captain, Saru is still only the first officer. They were on there way to Vulcan to pick up their new captain when the Enterprise showed up.