Here are three projections that result in models of spherical geometry. I think the stereographic model is closest to what you're looking for.
One possible stereographic projection projects from a pole onto the equatorial plane, giving a model that is analogous to the Poincaré disk model. (Projecting onto the equatorial plane identifies the equator with the unit circle. Other stereographic projections are scaled differently.)
Image credit to Leonid 2, CC-BY-SA 3.0.
This stereographic model has many desirable properties for geometry. It is conformal/angle-preserving, it can represent the entire sphere using the plane plus a point at infinity (cf. the Riemann sphere), and the great circles are the lines and circles that intersect the unit circle at two ends of a diameter. (This includes the unit circle itself.) Here's a nice visualization in GeoGebra. The stereographic projection is the only projection that represents all spherical circles as circles. (If we consider lines to be circles with infinite radius.)
This lab assignment from a 2005 Geometry for Teachers course at the University of Washington taught by James King has teachers explore great circles and spherical triangles under stereographic projection. You can find even more of his resources on this page.
We can think of the Poincaré disk model as the projection of one sheet of the unit hyperboloid of two sheets onto the equatorial plane, with the projection center being the opposite vertex.
Image credit to Selfstudier, CC0.
To facilitate comparison of the two models, we can restrict the stereographic model to only the projection of the hemisphere opposite the pole. This would make it a disk model. By identifying opposite points on the boundary, we could also consider it to be a model of elliptic geometry, which as this answer explains is distinct from spherical geometry.
Ian Stevenson developed a computer "microworld" for exploring these two models, and you can read more about it in his doctoral thesis or in this article. He attributes the stereographic model to Klein. He calls it the "hot-plate universe," and he calls the Poincaré disk model the "cold-plate universe." This language is from Jeremy Gray's Ideas of Space, and Gray reports getting it from Feynman. The idea is that, if we use a metal ruler as our metric, the ruler will grow as we move away from the origin in the stereographic model, and the ruler will shrink as we move away from the origin in the Poincaré disk model. Thus, paths far from the origin are shorter than they appear in the stereographic model, and they are longer than they appear in the Poincaré disk model.
You can read about pre-service teachers' investigation of the models in Stevenson's microworld in this article, this article, and this article.
Flattening the northern hemisphere in the z-direction onto the equatorial plane as you describe in the question is an orthographic projection. The orthographic projection is easy to visualize, but it isn’t conformal, only shows half the sphere, and crucially, an orthographic projection does not map great circles to circular arcs.
To see this, consider the orthographic projection of the globe shown. The edge of the projection is a circle representing the 30W and 150E meridians. The other meridians except for the central meridian (90E) are tangent to the outer circle at both poles, so they cannot be circular arcs. In general, an orthographic projection maps spherical circles to (possibly degenerate) ellipses. This article on spherical geometry by Lodge and Heawood discusses the drawing of spherical circles under orthographic and stereographic projections.
Image credit to Planiglobe, CC-BY 2.5.
A gnomonic projection projects from the center of the sphere onto a tangent plane. The resulting model of a hemisphere is analogous to the Klein disk model, which can be thought of as a projection of one sheet of the unit hyperboloid from the origin onto a plane tangent to the vertex. Like the Klein disk model, it represents geodesics as straight lines, but it also isn't conformal. Unlike the Klein disk model, it uses the entire plane. And to be a faithful model of elliptic geometry, we need the entire real projective plane.
Image credit to Marozols, CC-BY-SA 3.0.