Hating "politically correct bullshit" (or, as is more often the case when people complain about having to be PC, hating having to face consequences for deliberate rudeness) isn't the only reason why some people scorn the insistence on 'person first' language. Many of those who prefer not to use it are disabled themselves. There are some cases where there's grammatically no other option- there may be no adjective, or there is but it's a sesquepedalian monstrosity, or it's been co-opted into a slur (although I've seen efforts by various disabled people to reclaim 'gimp', 'cripple', 'spastic' , and the like for our own usage)... but most of the time, the choice isn't so black and white.
In everyday usage, we don't typically use "person with ___" phrasing exclusively except for strongly negative and/or medicalized traits. There are a few cases where the constraints of the English language, but you don't speak of, say, a "person with athleticism" or "person with female gender", and while a phrase like "person with blond hair" isn't unusual, it's freely interchangeable with "blond-haired person" or just plain "blond". It's only for diseases and disabilities that "person with" seems to be the only option in some people's minds.
And it's a bit stiff and unnatural-sounding, too.
The "with ___" is also thorny for some people who strongly identify with a disability/disability rights subculture, as well as those where what they are is deeply intertwined with who they are. Saying that they "have ___" or are a "person with ___" carries a bit of an implication that their ___ is something separate from who they are as a person, or else is only a tiny, unimportant part; if turning them into a non-disabled person would require a total rewiring of their brain, that's not necessarily true. And someone whose disability identity is an important part of how they self-identify may not approve of that sort of separation and minimization, either. He's not a person with a Roman Catholic upbringing, he's a Christian; she's not a person with deafness, she's Deaf; etc. It hurts to have your identity invalidated by having something important to you dismissed as if it were nothing.
The political correctness aspect can be a concern in and of itself, too. Respectful-sounding language isn't good for much if it's only surface-deep. There are plenty of people who mouth the 'right' words but don't actually respect the people they're talking to at all. I've personally been present for an ugly flame war or two on autism-related communities where some parents of 'children with autism' completely dismissed the opinions and advice of several autistic comm members, all the while insisting on referring to those members with 'person-first' language even though they knew the people in question identified as 'autistic' and didn't like the phrase. Anyone who disagreed with the specialists and parent support groups and autism 'charities' and so on was just WRONG and obviously didn't know what they're talking about, even if they were speaking from their own experience.
...finally, there's the argument that the sort of person who doesn't think of people with disabilities as real people isn't likely to shed that belief just by calling them as 'people' all the time.
There really is no current term or phrase that works universally. That's why try to make a point of deferring to people's wishes and calling them what they prefer to be called. The people who'd object aren't the ones whose opinion really matters, anyway.