How should I demonstrate a functional iOS application to a client?
Yes, there is a way to install apps on stock OS iOS devices, other than from Apple's App store or when tethered directly to the developers Mac.

An enrolled iOS Developer can sign an app with an Ad Hoc Distribution certificate and a mobileprovision file which includes the UDID's of the client's devices.  You can email this app and provision file to a client.  Then the client can install this app (and the mobileprovision files required for it to operate) using the iTunes app on their Mac or PC.  Or the client can download direct to their device from a (OTA) web page using HTTPS plus a manifest plist.

This is very commonly done for beta testing of iOS apps.

Ad Hoc provisioning expires at least once per year, so these apps are only temporarily useable.  A Developer is limited to no more than 100 UDIDs max total per year, so can only do this for a very limited number of clients.