I used to do a lot of transitioning between fast airplanes and small airplanes. To go fly a Lear 24 for a while, then jump back into an ag airplane was always an experience, but it was just as strange a transition going back. Different requirements, different ways of thinking and doing things.
In the more automated aircraft, there's definitely a loss of rudimentary stick and rudder skills among those who don't do any outside flying or engage in types of flying that can maintain those skills. I know some airline pilots who wouldn't think of doing any outside flying or training, and I know others who own their own light aircraft and do flying on the side. There's a difference in what could be expected from either group.
A regular part of our recurrent initial and recurrent training in a large Boeing used to be raw data, hand-flown NDB approaches. Presently, our internal policy is that "raw data" always includes the flight director, which came as a result of other changes regarding how we do approaches and our authorizations and capabilities. We still do the NDB in training, but now it has RNAV overlay, and doesn't resemble much of an NDB any more.
I personally know several individuals flying large airplanes who have stalled the aircraft inadvertently. One was a stall on departure in a 747-400, and came as a result of being new in the aircraft and not managing it properly. Unlike the 747-200 from which that pilot had come, nearly everything in the -400 is done through the Flight Management Computer. One becomes a typist and programmer, rather than an aviator. Unlike the classic, the airplane manages descent profiles, and nearly every other aspect of the operation. The crew only tells the airplane what they want it to do.
For those who don't seek proficiency outside of recurrent training, it's reasonable to believe that basic skills suffer. I know pilots who scarcely ever operate without the autopilot, and others who don't. I prefer to hand fly at least to FL270 or higher on every departure, and usually hand fly every other approach. When I'm at work, it's about the only actual flying I get to do; the rest is managing through the autopilot system. That said, the type of flying that's been done also makes a difference.
For those who are doing the same two or three destinations, under the same conditions, and all of it automated, it's possible that a number of skills may begin to atrophy. Not everyone does that, however, and not everyone that does that does only that.
Before making the generalization that the airline or corporate pilot can't fly, what of the private pilot that only gets an hour here or there each year? Everyone is different. I would say that flying a lot , every day, every week, every month, and getting regular, professional industry-standard recurrent training shouldn't be dismissed.
There are differences; in a light airplane, we practice full stalls. This isn't practical in a large four engine transport in most cases; we practice avoiding the stall in the first place recognition, and recovery. In a small light airplane while spraying a field, I do steep turns every 30 seconds at each end of the field, sometimes right to the buffet. In a large airplane, I'm limited on after takeoff to 15 degrees of bank because of my stall margins, until I have at least ten knots above my takeoff safety speed; even then I'm limited to 25 degrees of bank. Not nearly as dramatic, but mainly just different.
Flying an approach to landing at 630,000 lbs isn't just about basic stick and rudder skills, but about energy management. If following the localizer and the airlplane drifts to one side, then as soon as one corrects, one had better be correcting back, because it's a lot of mass to start moving in one direction, and it tend to keep going. It also means that if one gets slow, one may not be able to get fast again; it's not nearly as easy to recover as it might be in a light airplane. In that respect, it's a little like flying an approach in a Cessna 150 on a hot day at high altitude with two big adults; one needs to plan ahead because the airplane doesn't have the performance to save you from a bad decision. It's all relative.