I suspect this depends on the weather in your area - road cycling clubs near me often require mudguards for "winter club rides".
And yes, once your winter 'proper road bike' has mud guards, tyres with treads, and an all-weather-riding design people might start calling it an 'adventure bike' or 'gravel bike' instead.
If you see someone with mudguards on a road bike it's a sign they're a proper cyclist.
I don't think you're allowed to entry audax events without guards and you definitely won't be allowed on a club run without them. Getting a face full of oily road water for the guy in front is not exactly fun. Even small clip on "racing blades" will get you a sniffy look.
If you google images of something like Redbull Hardline pretty much all the bikes you see will have at least one front guard.
That's because no one wants a face full of mud at the wrong moment.