What if you need several polyfills? Drop them in too. You should always know what browser feature you are using. The same argument goes for using libraries. What if you need more than jQuery? Bring them in too. There are asset pipelines and/or pure front end packaging solutions like AMD to help you. What's the problem?
Why? If I just drop jQuery in and treat that as my API baseline, what do I lose? A little performance, a few hundred kilobytes' download (almost certainly cached from a CDN anyway). And it frees me up from remembering a bunch of corner cases and keeping track of a bunch of implementation details, letting me focus my attention on more important things.
The problem is that this shuts you out of a lot of hot, emerging markets that can be quite lucrative. You will never work for Google (as a front-end engineer) if you only know JQuery and not the native DOM APIs. Your mobile HTML5 apps will suck and will lose out to native counterparts in the app store. You're at a disadvantage in any competitive web market where people bounce from the website if it's too slow (there are more of these than you think).
If you're willing to limit yourself in this way, sure, go for JQuery. There is still a pretty booming market for Intranet apps or progressive-enhancement mostly-static sites where JQuery is just fine, and it provides a nice productivity boost for those use-cases.
The advantage of polyfills is that they give you flexibility and let you push the edge of technology today, and then they don't become obsolete tomorrow. As the web evolves you just get rid of the polyfill and you already know the latest & greatest of tomorrow. That lets you push into the emerging new markets that can be quite lucrative, at the expense of actually learning the (sometimes hard-to-use) APIs on the bleeding edge.
But for those aspiring to create big things another approach is to find a niche where you can offer something exclusive that people want. Thefacebook offered something that students would have put up with > 1s load times to see that they couldn't reliably get anywhere else. http://en.wikipedia.org/wiki/File:Thefacebook.png
You seem to be laying much of the "mediocre experience" on jQuery in this case. I would like to know why you feel this way. Seems to me that loading in polyfills for every browser inconsistency can lead you to the same problems you feel exist with jQuery.
I don't just care what my browser is doing, I care what all of them are doing; which often leads to using jQuery. Depends on the project and the number of expected browsers involved.
Polyfills are things you learn once and drop in once? So is jQuery.
What if the best experience to be provided suggests using jQuery? Would you use it?
All in all, to use or not use jQuery often depends on the project. If it fits, use it; if it doesn't, don't use it.
You silly goose!
But that isn't the case here. When something like jQuery comes along, hiding so many gory details, and has been tested to death both in development and in production use all over the world, we are all better off because we remove so much failure surface area from our code.