That's a nice thought, but there is no single USB-C headphone standard, so there are in fact different implementations and compatibility issues.
Consumer Reports:
The Google Pixel 2 Headphone Conundrum
In an online post, Google noted that only headphones compatible with USB Type-C Digital Audio will work with the Pixel 2. […] Analog Type-C headphones won’t produce any sound, Google says. Instead they will prompt a smartphone notification informing you that you need different headphones. […]
How do I know whether my headphones are compatible?
For now, Google recommends that you purchase headphones certified as “Made for Google.” This program, similar to Apple’s “Made for iPhone” program, confirms that the hardware is fully compatible with the Pixel 2. As of now, only three manufacturers offer headphones that have earned that distinction: AiAiAi, Master & Dynamic, and Libratone.
The Verge:
Buying USB-C earbuds for my Pixel 2 was incredibly annoying and expensive
Two weeks after starting my cheap Pixel 2 earbud search, I finally have a working pair — but they cost almost twice the amount I wanted to spend, and don’t feel very premium. If I lose or break them, it’ll cost me almost $50 and another 10-day wait. The next time I upgrade my phone, they may not be compatible. Even the Apple Store sells $29 Lightning EarPods. Google needs to do a lot better by its Pixel owners than a single $149 USB-C option. Even better, just give us back the damn headphone jack.