It's definitely a loose connection, usually near the hinge. Try pivoting the hinge a bit or squeezing the screen or base near the hinge and I bet it comes and goes. I had an old toshiba laptop do the same thing.
If you like Chrome but want a desktop, search for "chromebox" and take a look. Typically you will only get the 'box' and will have to add your own mouse/keyboard/monitor. Buy those used and you should be golden. The hardest part will be making sure the monitor has the correct connection or cable to connect to the box. HDMI, mini HDMI or micro HDMI are pretty popular these days but if the monitor only supports displayport or VGA or some other legacy connection then you'll need to adapt somehow. Adapters can often cause issues with supporting the proper resolutions and stuff, but it is usually fairly easy to find something cheapish that has modern connections.