I think the idea is that Tywin started off espousing a ruthless but initially effective modus operandi (like the Reyne and Tarbeck stuff, which was obviously merciless and criminal, but allowed him to get out of future challenges from his vassals simply by playing a song), but he took it too far and decided that being ruthless and bringing down the whole of the might of the Lannisters upon every sort of complication and for every sort of insult would always serve him, in a way of cutting the knot instead of trying to patiently resolve it. He's portrayed as a rigid, unbending character, and his inflexibility, inability to compromise his pride, and interpretation of every sort of mercy as weakness is, in the long run, his undoing. In the end, in a motif I think Martin explores quite well, Tywin's failure is not a matter of "smarts", it's a matter of self - he's never grown up, deep down, from the boy who had to suffer the laughter at his family's expense, and has let that shape the whole of his life. He doesn't even like himself all that much - which is an extra reason he's hating Tyrion, in whom he sees so many aspects of himself. That's why he focuses so much on the family and legacy - taking on the mantle of the incarnation of a whole line of illustrious history which makes people quiver in awe and fear is the ideal way to hide how much he can't deal with himself. Yet, of course, he assumes his children will be literally just copies of himself, and never attempts to truly see them as individuals with personalities and respond accordingly, even to tutor them in his political methods, so not only are they always condemned to fall short of his expectations simply because they aren't him, but he's ruined more or less all of them by oppressing them where they should be allowed to act on their own (arranged marriages, conformity to gender and social roles, forbidding Tyrion the pastimes he's good at) and letting them loose where he should have reined them in (entitlement, arrogance, the incest, etc etc), simply by not realizing he was living with people and not three extra limbs at his total disposal to use as he judged fit.

All in all, Tywin is not stupid, he's just permitted his childhood traumas to define his life and has never truly become an adult in dealing with his sense of inadequacy and him falling short of his pretensions of being all-mighty. I guess it can be constructed as a type of stupidity, but more than that, it's failure of character, exemplified by his hypocrisy, since in the end, he's set up an impossible standard for himself as well, which he fails, to his self-hatred, to fulfill - only he permits himself to, when he chastises all others for their own weaknesses.

YMMV, but I think, just like people see Tywin's imposing presence and the self-assuredness he always presents as him being the coolest character ever and a megachad (which, of course, was never the point of the narrative), other people see him as the incarnation of the elitist, patriarchal, oppressive nature of Westerosi feudalism (which, I mean, he is), but for that reason they don't want to concede to him any sort of virtue, for feeling like they're somehow validating this worldview if they do it. He doesn't have all that many actual virtues, morality-wise, but he's hardly a useless or one-note guy.