I agree with Cerbera. If I wanted to do this using a polygonal object, I would spend the time and model it properly.
There are ways, however, to cheat yourself out of your situation and still get those slices you want.
Option A): Using a polygon version similar to the on shown in your screenshots
For the sake of argument, let's assume that this polygon object is our only option. What I'm about to tell you may or may not work well depending on what issues are possibly hidden in your polygon object.
In order to prevent the knife tool from closing that geometry, make the Extrude object editable (create a flat object because that's easier to fix). Then use "Close polygon hole" to close all the holes in the globe. Once you've done that there's a better chance that the knife tool will make those cuts without causing too many issues. Aftwards, delete the polygons again where the holes should be. See the attached image 1. I used a random free vector spline I found on the internet to do this. It kind of worked. At least one half of the polygon object was usable after I had made the cuts. To fix the remaining issues I just added a couple of additional cuts, deleted the half I couldn't use and mirrored the good half over.
You'd still have to spend some additional time though to separate the slices in order to be able to animate those like the ones in that YouTube video. Is there a particular reason why you did it this way? If not, I would go for option B) below.
Option B): Voronoi Fracture
This is the fastest option, and the best one for a quick result. However, how well it works will depend on how screwed up the spline is that you downloaded. You may have to fix that first before you can do anything.
The one I downloaded to do this was a single spline/path that, however, consisted of several segments. I split those off and then used spline mask to create a proper single spline. I then used an Extrude object to create the 3D object.
I put the Extrude object in a Voronoi Fracture Object and used a parametric plane with a number of width segments and no height segments to create the slices. See image 2.
The geometry still is an ugly mess, but this was really quick, and I could even bevel the edges and get it to look nice in the render (image 3).
Cheers,
contrafibbularities