WebGL builds will sometimes fail to launch without error when using Build and Run

When working with a WebGL project in Unity, if the user uses the "Build and Run" feature to start the player, sometimes (2/5) the Editor will fail to launch the Player with no errors.

Reproduced in: 2019.1, 2019.2, 2019.3, 2020.1

Steps to reproduce:
- Create a new project in Unity with WebGL build support installed
- Switch the project platform to WebGL (File > Build Settings > WebGL > Switch Platform)
- Build and Run
- Observe the editor will build and launch the player
- Repeat the Build and Run steps, observing sometimes (2/5) the editor will fail to launch the WebGL player without any error other than provided by the browser trying to access the resource