The original question mixes terms that mean different things depending on your frame of reference.
In most aircraft, "trimmed for level flight" means constant airspeed for a given pitch/power setting. That means as a plane flies "level" around the earth, the aircraft will have a miniscule nose-down pitch rate to keep the airspeed constant, keeping it from climbing. This has nothing to do with the attitude indicator.
Now let's consider that attitude indicator... Any attitude system must have a reference plane. In spacecraft, the attitude system is arbitrary, and may or may not be set to precess or rotate in any or all axis to match that arbitrary reference plane. The reference could be relative to the sun, to stars outside our solar system, the earth, even the moon or any other object moving relative to the earth, sun, etc. In an aircraft, the reference plane for the attitude indicator is the earth directly below the aircraft and it is tangential to the surface of the earth (assuming a perfect uniform sphere). Some aircraft systems (usually ones based on a ring-laser gyro) are sensitive enough to detect the earth's natural wobble and rotation around the sun and must be specifically programmed to cancel that out, and are therefore programmed to precess as required to always reference the tangental plane directly below the aircraft. Cheaper attitude gyros simply precess to reference "down", ie. the relative vector of the local gravity. This is not a problem if the plane is usually in straight and level flight for most of the flight, but it can be a problem if the plane is maneuvering wildly or spends a lot of time in a constant turn. In those cases, the attitude gyro will precess to reference the aircraft's attitude as "level", even if it's in a turn. This is of course not good, which is why those types of gyros have caging knobs so you can re-cage them as required during flight.
The practical effect is that in an aircraft, if the pilot keeps the attitude gyro "level", he will not climb because the attitude gyro is generally set up to reference straight down, not reference an arbitrary absolute plane or attitude reference geometry. A vehicle that has a gyro that references an arbitrary but absolute geometry, would fly "straight" reference to that plane if the attitude reference was kept "level", and it would therefore climb (or dive, roll, yaw, etc) relative to the earth if it's absolute attitude was held constant without regard to the presence of the earth.
