Model disappears when viewing from up vector

Question

I'm sure there is a simple explanation for this behavior that makes mathematical good sense, but I need a way to look at my model along the up vector. I have tested it with the three unit vectors to confirm it is viewing along the up vector that causes my model to disappear, at least in the current configuration.

tldr; Model disappears when camera is along up vector, how do I fix it?

Edit: Also forgot to mention besides changing the up vector to confirm, I also confirmed this was the problem by looking off the up factor by a small amount like so, which works fine I could potentially use this fact, but on principle I would like to know why I can't view along the up vector and possibly correct it in a way that makes mathematical sense:

if (Keyboard.GetState().IsKeyDown(Keys.D3))
    _cameraPosition = new Vector3(0f, 1f, 0.000001f);

One Solution, as suggested by DMGregory. You can change the up vector when switching to the top view like so and that works, better than using float epsilon I suppose. Code below, but I am still not sure that this is the best way to accomplish this effect:

        if (Keyboard.GetState().IsKeyDown(Keys.D1))
        {
            _cameraUp = Vector3.UnitY;
            _cameraPosition = Vector3.UnitX;
        }

        if (Keyboard.GetState().IsKeyDown(Keys.D2))
        {
            _cameraUp = Vector3.UnitY;
            _cameraPosition = Vector3.UnitZ;
        }

        if (Keyboard.GetState().IsKeyDown(Keys.D3))
        {
            _cameraUp = -Vector3.UnitZ;
            _cameraPosition = Vector3.UnitY;
        }

Code to follow, monkey from blender is the idol:

public class MyGame : Game
{
    private GraphicsDeviceManager _graphics;

    private Model _idol;
    private Matrix _idolWorld;
    private Vector3 _idolPosition;

    private Vector3 _cameraPosition;
    private Vector3 _cameraUp;

    private float _width;
    private float _height;
    private float _near;
    private float _far;

    private Matrix _view;
    private Matrix _projection;

    public MyGame()
    {
        _graphics = new GraphicsDeviceManager(this);
        _graphics.PreferredBackBufferWidth = 1920;
        _graphics.PreferredBackBufferHeight = 1080;
        _graphics.ApplyChanges();
        Content.RootDirectory = "Content";
    }

    protected override void Initialize()
    {
        // TODO: Add your initialization logic here
        _width = (float)(_graphics.PreferredBackBufferWidth >> 1);
        _height = (float)(_graphics.PreferredBackBufferHeight >> 1);
        _near = -MathHelper.Max(_width, _height) - 1f;
        _far = -_near;

        _idolPosition = Vector3.Zero;
        _idolWorld = Matrix.Identity;

        _cameraPosition = new Vector3(0f, 0f, 1f);
        _cameraUp = Vector3.UnitY;

        _view = Matrix.CreateLookAt(_idolPosition + _cameraPosition, _idolPosition, _cameraUp);
        _projection = Matrix.CreateOrthographicOffCenter(-_width, _width, -_height, _height, _near, _far);

        base.Initialize();
    }

    protected override void LoadContent()
    {
        // TODO: use this.Content to load your game content here
        _idol = Content.Load<Model>("Idol");
    }

    protected override void UnloadContent()
    {
        // TODO: Unload any non ContentManager content here
    }

    protected override void Update(GameTime gameTime)
    {
        if (GamePad.GetState(PlayerIndex.One).Buttons.Back == ButtonState.Pressed || Keyboard.GetState().IsKeyDown(Keys.Escape))
            Exit();

        // TODO: Add your update logic here
        if (Keyboard.GetState().IsKeyDown(Keys.D1))
            _cameraPosition = new Vector3(1f, 0f, 0f);

        if (Keyboard.GetState().IsKeyDown(Keys.D2))
            _cameraPosition = new Vector3(0f, 0f, 1f);

        if (Keyboard.GetState().IsKeyDown(Keys.D3))
            _cameraPosition = new Vector3(0f, 1f, 0f);

        _view = Matrix.CreateLookAt(_idolPosition + _cameraPosition, _idolPosition, _cameraUp);

        base.Update(gameTime);
    }

    protected override void Draw(GameTime gameTime)
    {
        GraphicsDevice.Clear(Color.CornflowerBlue);

        // TODO: Add your drawing code here
        foreach (ModelMesh mesh in _idol.Meshes)
        {
            foreach (BasicEffect effect in mesh.Effects)
            {
                effect.EnableDefaultLighting();
                effect.World = _idolWorld;
                effect.View = _view;
                effect.Projection = _projection;
            }
            mesh.Draw();
        }

        base.Draw(gameTime);
    }
}

Edit for clabe45 (the model, near and far seem fine):

Answer 1

DMGregory suggested this answer. Thanks!

What he suggested seemed reasonable: https://en.wikipedia.org/wiki/Hairy_ball_theorem

To put this to the test, I created a bottom view perspective. The model did disappear as in accordance with the theorem, I should have at least two cow licks in this case.

Changing the up vector to accommodate the two unique cow licks turned out to be the answer:

    // side
    if (Keyboard.GetState().IsKeyDown(Keys.D1))
    {
        _cameraUp = Vector3.UnitY;
        _cameraPosition = Vector3.UnitX;
    }
    // front
    if (Keyboard.GetState().IsKeyDown(Keys.D2))
    {
        _cameraUp = Vector3.UnitY;
        _cameraPosition = Vector3.UnitZ;
    }
    // top
    if (Keyboard.GetState().IsKeyDown(Keys.D3))
    {
        _cameraUp = -Vector3.UnitZ;
        _cameraPosition = Vector3.UnitY;
    }
    // bottom
    if (Keyboard.GetState().IsKeyDown(Keys.D4))
    {
        _cameraUp = Vector3.UnitZ;
        _cameraPosition = -Vector3.UnitY;
    }