So in your example you're using a tower defense analogy. I'm not sure how the holy trinity is much different from a tower defense game. You have some long range tower and some short range towers and each is more effective against a certain type of enemy. So my question is, where do you make the distinction on what is 'more effective'? Say you have an enemy type (enemy A) you want to face that is best countered with a long range tower, but by happenstance you don't have any long range towers. Traditionally when this kind of setup occurs over a period of time players will start to develop the mentality that in order to fight enemy A they must always have a long range tower present in order to be successful. This change in mentality can be more or less severe based on how 'more effective' you make a long range tower counter enemy A.
Second this brings up how much solo vs. group play you want to encourage. Let's say you're a long range tower that is soloing and you encounter a group of enemy B, which is most effectively countered by short range towers. How hard is it going to be for you to progress if you mostly solo, or will it be a trivial matter because the game focuses on always being in a group? The same analogy can be applied to the famous elemental resists argument. You spec for ice, then part way through the game, you encounter a bunch of ice resistant or immune enemies. Should you be able to progress through on your own or do the designers feel that edging you into group play to overcome such encounters would be better?
The op I made was a result of sitting in a LoL queue for 15 minutes and getting kicked out of a group selection multiple times because someone would realize that no one on the team had chosen a tank character and would bail out of the selection. It didn't matter that the enemy team might not have had a tank player either, there was the potential that they could have had a tank and that our team would then be fighting at a disadvantage. Not many people enjoy fighting at a disadvantage, perceived or otherwise. The holy trinity exacerbates this problem. Having something setup to be 'more effective' could also mean to some players that conversely it's 'less effective' at other things and becomes a perceived weak class against things it doesn't counter. While providing job roles that are easily understood certainly motivates group play, ironically it can also get in the way if no one can fill a specific job. What happens to a soccer team when they can't field any forwards, or a basketball team without a center?
mouse clicking also makes strategic positioning difficult
I'm not sure what you mean by this. There are many mouse movement driven games out there that require strategic placement. I know it's not Risk, Disgaea, X-com or any other turn based system but that doesn't mean there isn't strategy between standing in front of or behind an enemy in an ARPG or whether you're in a good position to move out of a flamewave or not.