For one, WebKit is now forked, into WebKit and Blink. Both of them are rapidly diverging. And there are still enough people using Gecko and Trident (and now EdgeHTML) that you can't just target the intersection of WebKit and Blink and still have any reasonable kind of coverage.
For another, I think you fail to understand just how dominant and stagnant the browser market had become when IE was dominant. Not only one browser, but one version of one browser, had over 90% market share, and in controlled environments like corporate networks, effectively 100%. In that kind of environment, it was completely reasonable to just target that one browser.
Today, even if you add up all of the WebKit and Blink browsers based on the most generous market share counts, you won't break 70% market share, and there are even still some statistics that have IE/Trident at over 50% market share. So unless you're targeting a very specific market, like iPhone users (where yes, WebKit's share is 100% due to Apple's monopoly), it's very hard to find any demographic in which there is such a strong dominance as IE had back in the day.
Additionally, browsers and engines are actually being released at regular intervals these days, with new features and bugfixes. Firefox/Gecko and Chrome/Blink are on rolling 6 week release cycles, meaning that there's very little lag to getting new features and fixes out. Safari/WebKit is released less often, but once every year or two. There were 5 years between the release of IE 6 and IE 7, and IE 7 didn't even pass Acid2, it would be another couple of years until IE 8 did. That extremely long period of time in which one version of one engine was so dominant led to an awful lot of sites designing themselves exactly around the bugs and quirks of that one engine.
Couple that with the fact that standards committees are actually working on features that people want, and are making real progress. XHTML stagnated the standards process for a long time, working on a backwards-incompatible specification that did a lot of architecture astronautics on representing documents, when what people needed were backwards-compatible enhancements that supported web applications so they could roll things out incrementally as support became good enough. CSS got a bit mired in CSS 2, but CSS 2.1 allowed them to slim down the specification and get out something working, while CSS 3 allowed them to work on individual features separately, and thus be able to make progress without having to wait for everything in the world to be ready. Another big reason for IE's dominance was that it had proprietary features, like filters and ActiveX and webfont support, and there were no standards that covered similar ground. Now we have HTML5, a plethora of new APIs for interacting with the browser, a substantial number of CSS enhancements, highly competitive JavaScript engines that are capable of running full 3D engines, and so on.
And finally, WebKit, Blink, and Gecko are all open source. People can be involved in helping find and fix bugs, implement new features, port them to new platforms and devices. A big problem with IE was that it was proprietary, and ran on one platform. You could build a desktop browser that ran on Windows with your own chrome around it, but that was about it. Now, you want to make some cool new device that needs web rendering support? You want to experiment with interesting new ways to present web content? You now have several high-quality, open source options to choose from, which have enough market share that most sites have done the compatibility work to make sure they are compatible.
Sorry for the wall of text; I just wanted to make sure that you understand the history, and how very different the present situation is than the one in which the IE 6 monoculture flourished. It can be a bit worrying how two engines with a common ancestor have such dominance on mobile, WebKit and Blink. It's also worrying that Apple's monopoly prevents people from installing browsers with more up-to-date engines on their phones; I think it's an abuse of their control of the platform. But even so, they don't have the kind of market dominance even on smartphones that Microsoft did on desktops back in the early 2000s, so while it's worrying, it's nowhere near as bad.