Acer Swift 3 SF314-51-52KY Can't project to another screen

What is the exact Swift 3 model number, as plugging a monitor into the HDMI port should work 100% and connect this monitor to your Swift 3 laptop. Check the HDMI port if its not faulty or damaged, as and if this monitor and the same HDMI cable works with another laptop then its the Swift 3 HDMI port that is damaged and not working and the HDMI port might need replacing or the actual HDMI ports pins have dislodged and come lose from the mobo making it not work 100% and it might just needs resoldering, which should be done by a technician that has proper micro soldering tools, don't try to do this soldering job yourself as you will do more damage than fix anything as this is a specialist job.

No, the issue is that you don't have the correct display adapter driver installed. You shouldn't be on the Basic Display Adapter, that's only used when Windows doesn't find the right driver. If you can tell us which Swift 3 model you have we can point you at the right driver. It's usually on the same sticker as your serial number and looks something like SF3xx-xxx-xxxx.