Unfortunately it can be very difficult sometimes to get an Apple laptop to display video on external display when the built-in display has a problem. With the newer models it can get even more complicated since the Apple Silicon Macs will only display certain content only on the built-in Display....I have personally encountered this situation while supporting my organization's Macs.
With the older model MBAirs such as your 2015 model, I find that the Thunderbolt port tends to get very dirty...sometime even attracting liquid that may have splashed on the table. Inspect the Thunderbolt port closely to see if you notice any signs of dirt, corrosion (usually greenish white spots), signs of liquid residue, or even just oxidized contacts. If the port is only dirty or has oxidized contacts, then first use a bit of compressed air to clean out the dust. Then use a non-metallic item such as a wooden or plastic toothpick to scrape the Thunderbolt port's contacts. Just make sure to only scrape the contacts from back to front. Scraping back & forth or sideways may cause the contacts to become damaged since they tend to lift up easily.
Sometimes you may even need to use your display adapter.....inserting & removing it multiple times from the Thunderbolt port to help clean the contacts.
If you are not getting any signal on the external display even after cleaning the Thunderbolt port, then make sure to power cycle the external display. I find many of today's displays are quite finicky and just toggling the display inputs doesn't always do anything. Does your external display support any other video input connections? Sometimes HDMI is also finicky.
Next, if you do get a video signal on the external display, then you may need to use Clamshell mode in order to get video. While you should get an extended desktop when booting normally, it can be hard to enable display mirroring since many times that option may not work until after successfully logging into macOS. Clamshell mode requires the laptop to be connected to the power adapter and a wired external USB keyboard & mouse are also needed. When all these things are connected to an Apple laptop, then closing the lid (aka built-in display) should enable Clamshell mode where all video is sent to the external display. Many times I try to close the lid of the laptop immediately after hearing the startup chime so the very early boot process can show on the external display.
