I agree with this. I think it's normal and okay to have mechanics that naturally counter each other, but when Riot goes out of their way to implement second- or even third-hand counters, whether they be on champions or itemization, that's when things get messy. This is also one of those situations where I feel player perception isn't necessarily the most helpful either: everyone wants Grievous Wounds when healing is excessive, for example, but now that pretty much anyone can easily access GW and cut the enemy's healing by almost half at all times, healing now has to be balanced around that, and is consequently too strong. This is why Mundo was buffed to regen 100% of his max health, for example.
In general, counters to counters that are too widely available tend to drive up whichever mechanic they are designed to counter, since that mechanic then has to be strong enough to survive being constantly countered. This is also why League has a problem with tanks, because there is so much anti-tank damage in the game that tanks need to be buffed in compensation. Unfortunately, those buffs have frequently been in the form of damage, rather than durability, which has harmed the class's identity. In each case, though, these counters generate an arms race, one where both side are driven to creep up in power, to the detriment of anyone who hasn't caught up yet.
In the case of champions, I feel there's been a similar problem, in that many recent champions have been designed to counter their own counters. Whereas older champions generally have some situations where they'll always suffer, many more modern champions have been loaded with enough mechanics to have a response to practically any situation. Contrast Ashe, for example, who has super-clear strengths and weaknesses, and who gets punished when she messes up, to someone like Zoe or Camille, both of whom can make plenty of mistakes and still have built-in recovery options. Personally, of the three designs, I'd say Ashe is the most successful by far, yet this isn't the design philosophy that is being followed anymore, at least not with the majority of reworks and new champions (it's also worth mentioning that Ashe's rework wasn't all that long ago).
In a League where everyone could recover from their mistakes, I think champions like Yasuo, Zoe, the new Akali, etc. would all be more acceptable, because everyone would get to counter everyone else, but that is simply not the case in League. Some champions can punish others hard, and some champions can recover pretty well, but not everyone can do both. As such, when you add champions designed to punish enemies extra-hard with flashy combos, but also give them really strong options for when they mess up, those designs feel unfair, no matter how much goes into balancing them. As with many parts of its design, League is stuck halfway in a transition between its old state and a new one, and the resulting impact on gameplay isn't great.