OpenGL point sprites not always rendered front to back - opengl

Question

I'm working on a game engine with LWJGL3 in which all objects are point sprites. It uses an orthographic camera and I wrote a vertex shader that calculates the 3D position of each sprite (which also causes the fish-eye lens effect). I calculate the distance to the camera and use that value as the depth value for each point sprite. The data for these sprites is stored in chunks of 16x16 in VBOs.
The issue that I'm having is that the sprites are not always rendered front to back. When looking away from the origin, the depth testing works as intended, but when looking in the direction of the origin, sprites are rendered from back to front which causes a big performance drop.
This might seem like depth testing is not enabled, but when I disable depth testing, sprites in the back are drawn on top of the ones in front, so that is not the case.
Here's the full vertex shader:
#version 330 core
#define M_PI 3.1415926535897932384626433832795
uniform mat4 camRotMat; // Virtual camera rotation
uniform vec3 camPos; // Virtual camera position
uniform vec2 fov; // Virtual camera field of view
uniform vec2 screen; // Screen size (pixels)
in vec3 pos;
out vec4 vColor;
void main() {
// Compute distance and rotated delta position to camera
float dist = distance(pos, camPos);
vec3 dXYZ = (camRotMat * vec4(camPos - pos, 0)).xyz;
// Compute angles of this 3D position relative center of camera FOV
// Distance is never negative, so negate it manually when behind camera
vec2 rla = vec2(atan(dXYZ.x, length(dXYZ.yz)),
atan(dXYZ.z, length(dXYZ.xy) * sign(-dXYZ.y)));
// Find sprite size and coordinates of the center on the screen
float size = screen.y / dist * 2; // Sprites become smaller based on their distance
vec2 px = -rla / fov * 2; // Find pixel position on screen of this object
// Output
vColor = vec4((1 - (dist * dist) / (64 * 64)) + 0.5); // Brightness
gl_Position = vec4(px, dist / 1000, 1.0); // Position on the screen
gl_PointSize = size; // Sprite size
}
In the first image, you can see how the game normally looks. In the second one, I've disabled alpha-testing, so you can see sprites are rendered front to back. But in the third image, when looking in the direction of the origin, sprites are being drawn back to front.
Edit:
I am almost 100% certain the depth value is set correctly. The size of the sprites is directly linked to the distance, and they resize correctly when moving around. I also set the color to be brighter when the distance is lower, which works as expected.
I also set the following flags (and ofcourse clear the frame and depth buffer):
GL11.glEnable(GL11.GL_DEPTH_TEST);
GL11.glDepthFunc(GL11.GL_LEQUAL);
Edit2:
Here's a gif of what it looks like when you rotate around: https://i.imgur.com/v4iWe9p.gifv
Edit3:
I think I misunderstood how depth testing works. Here is a video of how the sprites are drawn over time: https://www.youtube.com/watch?v=KgORzkM9U2w
That explains the initial problem, so now I just need to find a way to render them in a different order depending on the camera rotation.