Avoid considering a horizontal tail with zero downforce, it is so incredibly rare in certified airplanes, it is not worth consideration, and really fusses up the logic path. The tail always exerts some downforce - more downforce for forward C of G, less for aft C of G. The difference is trimmed out by the pilot to bring the pitch control force to zero (but there is still tail downforce, it's just trimmed to no control force for pilot comfort). For economy cruise flight, and aft C of G is preferred, because less tail downforce is required, so less drag.
If there were no tail downforce, the airplane would not behave as expected (and required for certification) in stall recovery. This is a prime element of determining the aft C of G limit - there still must be enough tail downforce, that the nose wants to drop when down elevator is applied. During spin testing, I have flown a few types, where spin recovery at the aft C of G limit requires that full nose down elevator be applied and held during the recovery - you know that you're at that plane's aft C of G limit then!
Certification requires that a pull always be required when airspeed is slowing toward the stall. This pull could be trimmed out if the pilot desires, but is the stability indicator to the pilot that they are approaching the stall. I have flown two types where under certain configurations a pull was not required to slow, but rather at a certain [slowing] airspeed, the pitch control force went to zero, and then a restrained push was required to slow more. This is a pitch unstable airplane. The two examples I have flown which had this characteristic were the turbine DC-3 and the Siai Marchetti 1019. It is very un nerving! Modern certified airplanes either do not have this characteristic, or have artificial stall barriers to simply keep you away from this regime of flight.
is there a parallel with taildraggers risking groundlooping, because their CG is behind the main wheels?
I was initially thinking to say no to this, but upon further reflection, it's not a full no. In the yaw axis, a taildragger is stable during ground roll until the fin stops being effective, then a groundloop is possible if the pilot is inattentive to maintaining heading with rudder. This does have similarities to the pitch scenario, when the horizontal tail stops being effective, and the nose drops. So yes, kinda like a groundloop in flight in the pitch axis, other than it is an intended design characteristic for safety, and automaticcally corrects itself when airspeed returns (which a groundlooping taildragger generally would too, if the pilot increased speed during the groundloop, to make the fin effective again. Any stalled tail surface ceases to be effective in maintaining stability in that axis.