I wouldn't try to convince them with words but by letting them hear gear I have experienced and let them decide for themselves. I would have them listen to the Rode NTH-100 as a comparison to budget gear to start. Then I would move on to the Austrian Audio Hi-X60, Sennheiser HD8xx, Audeze LCD-2 Closed, Dan Clark Ether C, Rosson RAD-0, Audeze LCD-5. I would use the Mytek Liberty DAC ii and HeadAmp GS-X Mini. I would play music from my reference playlist: https://www.thehonestaudiophile.com/post/does-your-music-preference-impact-your-sound-impressions

Staying in the $200 price range is a tough one, especially for closed back options.