but this breaks the entire premise of the agent. If my emails are fed in as data, can the agent act on them or not? If someone sends an email that requests a calendar invite, the agent should be able to follow that instruction, even if it's in the data field.

It would still be able to use values extracted from the data as arguments to it's tools, so it could still accept that calendar invite. For better and worse; as the sibling points out, this means certain attacks are still possible if the data can be contaminated.