The acceleration of gravity is not always assigned a negative value. It all depends on your frame of reference and direction of motion.
If you throw an object upward and designate that direction as positive, then gravity is pulling the object downward, slowing its upward velocity. Therefore, the effect of gravity is working against the motion and has a negative value.
However, if you are on top of a building and drop an object, then the effect of gravity is working in the same direction as the motion of the object and is assigned a positive value.
The reason gravitational potential is always designated as a negative number is that this sets up a frame of reference to visualize the question in. The center of the earth is designated as "down", which lets us define movements away from the earth as up, and so on. The sign in this case indicates direction along the up/down, or Z, axis of motion.
Picture this: you are on a rocket looking back at the earth. You can see a person standing near the North Pole, and another person standing near the South Pole. Both of them are holding a ball, and they simultaneously drop their balls. What do you see? The two people who dropped the balls will each see their ball moving "down", but to you the balls will appear to you to be moving toward each other.
What you have touched on here is a piece of the Theory of Relativity. Einstein realized that all motion is relative, and you can only talk about it meaningfully if you have a framework to compare the motion against. Usually we use the Earth as a fixed reference point for motion, treating it mathematically as an unmoving platform beneath us. Calling gravity negative means that you should assign upward motions a positive value, so that the math comes out correctly.