You're going to get all sorts of confused by this thread.
Why not just grab a local CFI and explain that you've seen all those goofy YouTube videos of negative G pushes and that you'd like to experience a mild safe one in a real airplane. You know there are some risks involved and you don't want to accidentally do anything dumb or dangerous.
Bonus: Find the CFI who'll do spin training and unusual attitudes. If they recommend the use of something aerobatic-rated, fine... but still have them show you mild maneuvering in the Skyhawk.
Also, even though you're not going for the rating, any CFI can demonstrate the maneuvers required of the Commercial rating, for example, which are good aircraft control maneuvers and very benign. They won't feel benign if all you've ever done is pattern work. They will however open your eyes to the three-dimensions we operate in. Look at the books and the descriptions. Download a copy of the Commercial PTS and envision in a chair how the controls would have to move to do those maneuvers.
After you've seen and felt it, it's a lot easier to understand.
Airplanes only fly square patterns because we make them. Climbing and descending circles in the sky are fun. Vertical arcs while coordinating throttle and elevator inputs to hold one airspeed are good practice. So called "Dutch rolls" where the heading of the aircraft stays constant in one direction but you rock the wings from side to side while using coordinated rudder inputs to stay pointed straight ahead are also easy and a hoot.
A Skyhawk isn't suited for the "tumbling mirth" of the famous aviation poem,
but it doesn't have to be a "straight and level all the time" airplane either.
Have a CFI with grey hair show you how to play safely if the ones with brown hair won't.
Remember above all the point of learning aircraft control feel is so you can use those motor skills to get yourself out of trouble and also so you can give most of your passengers a smooth comfortable ride. If one asks for the more aggressive maneuvers, a simple steep turn may fit ther desire for "pulling Gs" just fine.
And check on their condition often, or you'll be scrubbing up the barf out of the carpet.