You are all kind of right,

"climb with power, and you control airspeed with the elevator, anyone with a pilots license can tell you that"

This will in fact happen under certain conditions such as MCA or during a landing when you are on "the back side of the power curve". Like our models, during normal flight conditions, the elevator is the primary control for altitude.

Sooo... if you fly around in MCA (Minimal controllable airspeed) up thrust might work, otherwise add down thrust like all the pattern & IMAC trimming charts will tell you to do for this symptom. Wing to horizontal stabilizer incidence is a good place to start, check that first, and if it's ok add down thrust.