Instead of thinking of "local flatness at each point" as an intuitive definition of a manifold, you can think of it in this, slightly more accurate way:
A manifold is some object that has the property that at each point on the surface of this object, you can take some disc cutout from a very stretchy material so that you can place the centre of this stretchy disc on the chosen point without cutting or tearing the disc, and deform the disc to fit "nicely" on the surface of the object, without any parts of this disc not being stuck to the surface of the object, and such that the disc doesn't overlap itself anywhere
This is probably the closest you will get to the formal definition of a manifold, and it allows you to see that the pyramid is a manifold, since even at the sharp points of the pyramid, you can simply place the stretchy disc, and stick it smoothly to the faces that surround that point, and since the disc is so stretchy there are no creases and it doesn't need to get torn. However if you try to do this with the two pyramids with their points stuck together, the only way to even get the centers of the disc and this meeting point to align is to first fold the disc in half, an then bring the centre of the disc to the meeting point, but then if you wanted to smooth out the disc around this meeting point, you would need to wrap it around the pyramids, and you might be able to imagine why no matter how small you choose the disc, it would always overlap with itself if you get close enough to the meeting point. This of course does not constitute a proof since we don't know that there aren't other ways to make the centres meet or smooth out a potential disc, but it should at least give a bit more intuition.
I'll also give a brief explanation of why this intuitive definition is a good starting point for someone that isn't familiar enough with Topology:
The formal definition of a manifold is that it is a set $S$ together with a topology on $S$ such that every point in $S$ has a neighbourhood which is homeomorphic to some open subset of $\mathbb{R}^n$.
For your purposes it suffices to think of a topology on a set as a way to determine the closeness of points in $S$ to each other, which induces some sort of "geometric" structure on the set. A neighbourhood of point is then just some subset of $S$ containing that point, which is "localised" around that point with respect to this "closeness" that we mentioned earlier. In our pyramid example, the neighbourhoods of the pointy tip of the pyramid were all those points on the surface of the pyramid within a certain distance of the pointy tip. The open subset of $\mathbb{R}^n$ was the disc, i.e. a subset of $\mathbb{R}^2$, and the condition that The neighborhood be homeomorphic to this disc was captured in everything else: The disc being forced to be smoothed out on the surface corresponds to the definition of homeomorphic requiring a function from the disc to the neighborhood of the pointy tip (The function has to map every point in the disc somewhere so the disc cannot stick out, or not touch the surface anywhere). The description of the disc being stretchy, but not being allowed to tear corresponds to the requirement that the function from the disc to the neighborhood should be continuous. And the requirement that the disc should not overlap anywhere corresponds to requiring that the function sending the disc to the neighborhood should be a bijection. There was just one requirement that we did not build into the intuitive definition, which is that the inverse function, sending the neighborhood to the disc, should also be continuous, but I couldn't think of a nice visual interpretation of this.
