The most common killer of inadequately trained pilots who fly into clouds is a spiral dive, caused by the aircraft entering a steep bank without the pilot realizing it or understanding the direction of turn. Eventually the aircraft is destroyed due overspeed or excess G-load.
A magnetic compass is generally useless whenever an aircraft is banked and turning, so forget about trying to use that to keep the wings level, except in some very particular circumstances. Hence the importance of a turn rate indicator or attitude indicator.
Many pilots have flown sailplanes in clouds with a turn rate indicator as the only gyroscopic instrument present. Charles Lindbergh did a great deal of cloud-flying during his 1927 Atlantic crossing, and a turn rate indicator was his only gyroscopic instrument. So the general answer to your question is "yes, a turn rate indicator is a valid backup for an attitude indicator". But read on for more--
The turn rate indicator can tell you which direction you are banked, and can give you a clue as to how steeply you are banked. However, with no attitude indicator, it can be challenging to manage the aircraft's pitch attitude. Lots of practice in safe conditions is needed. Flying with the turn rate indicator as the only available gyroscopic instrument is called "partial panel" flying, and pilots practice this in safe conditions outside of the clouds by covering up all the other gyroscopic instruments and donning a blind flying hood to block the view out the windows.
So how is pitch attitude controlled in "partial panel" flying?
If the airspeed is changing, that is a valuable cue to pitch attitude. Inside a strong updraft or downdraft, you can't rely on the vertical speed indicator, or changes in the altimeter reading, to tell you the aircraft's pitch attitude. Therefore you could argue that the airspeed indicator and the turn rate indicator are best pair of instruments to serve as a backup for attitude indicator, if you can only choose two and need to stay right-side-up in turbulence. Of course the heading indicator also gives some clue of the turn rate and therefore the bank angle, and it also gives you heading, so that's a very nice instrument to have as well. In some conditions, the heading indicator plus the airspeed indicator might be the best choice for a pair of instruments to stand in for the attitude indicator.
The relationship between airspeed, pitch attitude, and vertical speed will depend on aircraft performance as well as updrafts and downdrafts. A pilot flying a fast jet might have better results relying primarily on the altimeter for control of pitch attitude even in significant turbulence, while a pilot flying a light plane in the same conditions might have better results relying primarily on the airspeed indicator for control of pitch attitude.
Don't forget that the airspeed indicator can fail due to ice in the pitot tube, unless the pitot tube is heated. That would be a bad thing to happen if your life depended on it.
If you are trying to use ONLY the turn rate indicator (or ONLY the heading indicator) to substitute for the attitude indicator-- i.e. if you are trying to fly using no pitch information of any kind, not even from the airspeed indicator or altimeter-- then you had better be flying a very pitch-stable, well-trimmed aircraft. This has been done, but it has not always ended well.