What are the pros? Smaller requests are only going to have a very small effect on speed or bytes used.
Also, some privacy improvements by having less data for browser fingerprinting.
Also, if we'd consider dropping User-Agent header altogether, less ways to detect an UA, so a bit harder to show "sorry, your browser in unsupported, this site is IE^H^H Chrome-only"
Obviously, just UA header isn't even remotely sufficient for any of those reasons, but would be a good start.