Can Someone Explain the Philosophy Behind Bruisers/Juggs?

Disclaimer: This isn't a QQ thread. Generalizations are generalizations; outliers are outliers. Ex: talking about mage mobility, Syndra and Zyra are typical of the generalization while LeBlanc and Ahri are outliers.

I'm genuinely trying to understand the philosophy behind the champ balance of guys like these:

There's a lot of variation there, but the common theme is that they can all kill squishies while building tank. They're also either very fast, have severe slows, or dashes (or a combination). I recently started playing and played a lot of before he got hit hard this patch. I would put them into this category as well.

I thought that playing would help me understand better, but it ended up reinforcing the feeling that bruisers are simply the strongest class in league. The only time I remotely struggled with in an equal 1v1 was, of course, against a Gnar. In fact, there were multiple times that I caught somebody in a combo and killed them by accident. In other words, I had built so little offense that it didn't even occur to me that the enemy might possibly die. I've literally blurted out "whoa!" or "holy shit!" on occasion after vaporizing an opponent.

Honestly, I just don't understand why tanky fighters do so much damage while having so much effective HP. Mages and assassins typically have around 1/3-1/5 of these guys' EHP, but mages and assassins aren't putting out 3-5x the damage in a combo given equal income.

This comes down to the weaknesses of the different classes. Squishies have to kite, and bruisers have to catch, right? But squishies have to kite or die, and bruisers have to catch or....what? Beyond that, kiting is inherently more mechanically demanding than chasing something down. Landing skillshots on the run, stutterstepping, managing cooldowns, and animation lock are all more difficult than chasing at a high move speed and using your speed boosts and gapclosers on cooldown. In short, the kite/chase paradigm puts squishies at a risk/reward disadvantage while also being much harder to execute in general. Why is balance tipped so heavily towards bruisers? If you disagree, explain why and based on what experience.

Useless Backstory: I started feeling there was an inherent power imbalance between tanky champs and squishy champs around the Devourer patch. I was languishing in low Silver and got tired of every enemy champ being a threat. After having played only mages (and Teemo), I picked up Singed. With all of zero experience playing either a melee or a tank, I went from Silver 4 to Gold 3 series in exactly twenty days and had a Singed win rate of around 70%.

Having reached my goal of Gold in my first season, I decided to branch out and learn two of the coolest champs that I sucked with: LeBlanc and Katarina. Turns out, it's a lot easier to whine about LeBlanc than to play her (ha!). I stuck with Katarina and did really well with her until preseason. Now I'm back down to Silver 2 after, again, playing mages (and one assassin).

So here I am again, stuck in Silver and resigning myself to the fact that I'm going to need to pick up another bruiser to climb back to where I left off with Singed. Not to say that mages aren't capable, just that I'm not good enough with them. But it makes me wonder how I can do worse with mages I've played for hundreds of games than bruisers I'm just picking up for the first time. Maybe I'm just a bruiser prodigy (not likely), or maybe bruisers just have too many inherent advantages at equal skill level (more likely). Why is this the case (if it is), and what is the design philosophy that maintains a balance discrepancy that feels pretty obvious?

TL;DR: What the design philosophy is behind bruisers doing current damage levels while being so difficult to kill. It seem they have the upper hand against the most classes while being the hardest to kill for the most classes. The damage to EHP ratios seem like they're out of proportion, but why?