The problem is that no "intuitive" explanation can capture what gravity is actually about, because if it could, then general relativity should itself be intuitive.
The rubber-sheet analogy is in my view not a totally misleading analogy for what it wants to show (namely that masses curve spacetime), but it tackles the wrong problem - the main problem being that our intuition about how motion happens in curved space is missing.
How can you make it right? Maybe combining a few methods can help?
Start with Newton's first axiom. If there is no force, the motion of an object remains in constant velocity. This is something many people can grasp nowadays (it wasn't intuitive before Newton though). Also, it's something that remains true in general relativity - if we take velocity in curved space.
Now, let's keep Newton's axiom and make space curved. What does that mean? You need to show how in a curved space, straight lines can actually look curved. Some pictures/videos giving some intuition about this are:
https://www.youtube.com/watch?v=jlTVIMOix3I (one of many similar videos actually "curving space" and putting straight motions on curved space to show what this means for someone who perceives the curved space as flat). The video particularly clarifies how something that seems to not be moving can be accelerated due to the curvature of spacetime: velocity is not a concept just in space, but the underlying concept is space+time.
Take flight routes over the atlantic: If you compare how they look like on a regular mercartor-projected map of the earth (curved and weird: why should this be a good route?) in comparison to the globe (the paths are the shortest paths - they are also actually "straight"), this gives you some more intuition.
Both analogies have their own drawbacks (the second one for example doesn't really exhibit unconstrained motion: planes are of course subject to earth's gravity), but maybe put together they do give an intuition of what is actually going on.
Now you have an intuitive feeling of what space curvature can mean for the motion of an object and how a curved space (first video) can actually make a baseball that I throw up fall down again, although no force acts on it and it moves in a straight line at constant velocity.
However, what general relativity is really about is that it tells us that masses cause space curvature. This is something that is (to some extent, but I don't know a much better analogy) illustrated by the rubber sheet analogy: If I put some mass on a flat space (rubber), the space is no longer flat (in the usual sense). The higher the mass, the deeper the sink becomes, which gives an intuition that there is an increase in curvature if we approach an object. What this means for the motion is illustrated in the videos about motion in curved space.
If you don't know what curvature means for the motion of a particle, the analogy of the rubber sheet is totally confusing. If you do know it, I believe it does have some merits - but maybe if you have intuition of what motion means in curved space, you don't need an intuition of how masses curve space...