My answer is obviously biased towards the public courses I was usually tasked with designing, but even when I talked with Tour pros or low handicap players, they never thought an unmakeable putt was a good thing. For that matter, course managers, concerned with the pace of play really Blanche at suggesting those kinds of wild contours. Supers usually don't like them either, since steep slopes tend to dry out faster, and the transition area between flattish and steepish can be hard to mow well.
Really, outside of the 1500 participants here, who seem split about 50-50, I have rarely heard someone advocate for a green of impossible putts. Outside this site, the percentage that would favor even the occasional impossible putt is probably less than 2%. That said, it is often possible to end up with one from somewhere to somewhere else on a green even if the gca didn't intend it, which is why most gca's (I think) have tended to make nearly all of the interior of most greens under 3% (i.e. reasonably puttable) knowing that a few areas tend to end up more difficult by construction accident or change from the plans. (Some can end up too flat with continuous dragging before seeding, too)
In the case cited in the OP by Matt, while I never did it, many gca's (usually associated with Tour pros) advocate for the lower tier of a 2 tier green to actually have a low ridge on the front to prevent "de-greening" from the upper tier. Unless the hole is so far uphill you can't see the green and/or the approach area so steep that de-greened putts (or high spin approaches) might run or backup down the hill to many yards in front of the green, I prefer to simply flatten the front to an under 2% slope to minimize those chances.
BTW, I recall reading on this site many years ago, that any putt coming over a 1-2 foot rise from above has a minimum run out of 14 feet. Usually, a super will keep pins at least 10 feet from the top side of a stair step rise in the green, but sometimes they need to set it within 10 feet on the lower level just to spread out wear. Thus, even the most delicate putt from above will run at least 4 feet past the hole, probably more. I forget who or when that was posted, but it was one of the more useful facts I have gained in years on this site.
I understand the objection to NOT having impossible putts for those who don't mind them here - If you design to eliminate them, green contours get pretty bland, although, for high play courses, like it or not, that is probably the most practical way to go, or at least, have some of those cup areas under 3% in each area of the green. And for budget courses, who just can't build endlessly large USGA or similar greens, limited green space doesn't reduce the number of pin locations required, explaining why so few public courses have fairly bland contours.
I know Tom Doak and I had a brief exchange on this in another thread, where he didn't think the wild contours punished average golfers all that much. Maybe not, and in theory, green contours favor the creative golfer over distance, obviously. My thoughts ran to just how much the average golfer would miss a putt on steeply contoured hole locations, and how many times that would lead to a 3 putt....slowing play, and not giving much joy. What is the over under of a C player making the comeback 10 footer? According to Broadie, 20% chance of making it, and a 2% chance of 3 putting, so basically, guaranteeing 2 more puts or a bit less from 10 feet for an almost guaranteed 3 putt. The actual percent made might depend, I think, on whether the initial 10 foot miss was caused by one big break away from the hole, and whether the cup area itself was reasonably flat in the last 3 feet around the hole.
Which reminds me, most golfers would prefer an architect design greens that give impossible putts to their opponent.....but never to them......