Hhhmmm ... that got some reactions ...
So...Capt Tullamraine,
.. actually wearing my Consulting Engineer's hat at the moment.
what would be the problems a pilot would face flying a "conventional" aircraft with a near zero pitch force gradient?
A very difficult aircraft to fly. Unacceptably low stick force/g. Result is that the pilot is set up to overstress the aircraft .. and maybe rip the wings off. Not a good life strategy.
What if the 737 auto trimmed for zero stick force?
You are confusing two things here, unfortunately. Long stab is about STARTING from a trimmed on-speed condition and then looking at the stick forces needed to maintain off-trim speeds.
(a) if you use the stick to fly slower, you MUST have a pull force. Conversely, if you fly faster, you MUST have a push force.
(b) if you release the stick force steadily the aircraft MUST resume a speed somewhere near the original trim speed
(c) the gradient has to be acceptable and MUST NOT reverse. Otherwise one needs to look at SAS kit such as you see on some light aircraft turboprop conversions. Generally, this is to address a stick force problem during the missed approach.
I thought it was the aircrafts tendancy to return to its current state if disturbed.
Common misconception associated with poorly executed basic theory training using the "ball-in-a-teacup" analogy. Nothing wrong with the analogy .. just that it is explained incorrectly most times. Your statement is a part of the story only .. see (b), above.
If you are interested, the current words for FAR are at
(a)
FAR 25.173 for the basic reg,
(b)
FAR 25.175 for some specifics, and
(c)
AC 25-7A for the testing.
There are a few matters where the certification bits get a little smudged by the time the pilot training system gets to it ... leading to the odd misconception amongst the piloting fraternity.