If Texture is "checked" in the Orth menu, all three Ortho views will display the texture... unless you've also selected Wireframe, in which case none of them will.
These settings are made in the Ortho menu selected at the top of the interface, not the Ortho submenu within each of the 3 Ortho windows. These Ortho submenus are only for setting the Background Image in each of those windows.
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Above is an Ortho window with texture and Wireframe BOTH checked.... but no texture appears, because it cannot while displaying Wireframe.
Above is an Ortho window with Texture checked, and Wireframe UNchecked. The texture appears normally.
Above is an Ortho window with Texture and Wireframe BOTH UNchecked. You see the surface and no texture.
The 3D view has its own menu, where you can also choose to display Texture and/or Wireframe. But these two window types (Ortho and 3D) are independent of each other as far as what is or isn't displayed in them.
Also, if the texture shows in X-Plane, then it's evident that a texture IS applied to it.