Could a helicopter fly like a plane this way?
You've described a tail-sitter aircraft. Instead of a helicopter that tilts sideways to cruise, it's a airplane that can sit on its tail and uses thrust to take off vertically, before transitioning to fixed-wing horizontal flight. This makes more sense since a plane spends more time in cruise than takeoff.

They were first demonstrated by Nazi Germany during WWII. A major benefit over the tiltrotor or tiltwing is that you don't need the bearings that can support the thrust of the engines or lift of the wings. You can also use a single centerline engine. Existing designs use contra-rotating propellers or jet thrust vectoring for anti-torque instead of the helicopter tail rotor.

The idea has been revived recently for UAVs, which don't have the occupant comfort problem:


Aerovironment SkyTote (US Government photo)