(Disclaimer- I am no expert, but have a descent amount of experience with sound systems)
Well it sounds to me like you've already solved your sound problem.
But if you wanted to keep the factory head unit and create some awesome sound, this it what I would try!
(and will be doing shortly!)
Your cutoff is from not having enough power from your factory unit to power the speakers at that high of a level. As for swapping out your tweeters, I would go ahead and say leave them as they produce the "tweets" or the highs in the music.
First I would replace all 4 door speakers with high quality speakers capable of producing high quality sound. Your looking for something that can handle
a high continuous power load, or "watts". Along with built in tweeter's to give you that "high Range" in sound. Here is a link to some.
Car Speakers Review 2013 | Best Car Sound | Compare Best Car Speakers - TopTenREVIEWS
And a link to replacing them.
Hyundai Sonata Door Panel Removal & Speaker Replacement Guide - 2011, 2012, 2013 Model Years - Picture Illustrated Instructions
After that I would run a door speaker amplifier to all 4 door speakers. This should make your bass cutoff disappear. As well as make your speakers sound EXCEPTIONALLY clear to the ear. This would take care of handeling the "mids" in the sound, as well as make the high's sound much clearer.
Here's a link to a reference as to what to look at.
Best Car Amplifiers – Compare Reviews and Rankings | Bestcovery
Then to hand the bass. In order to keep the factory unit you will need to get an RCA convertor. Or this.
Scosche SLC4 Four-channel line output converter at Crutchfield.com
A scosche convertor. It converts from stock speaker wire to RCA output so you can run your RCA jacks to your sub.
I've written a DIY on this on this forum
DIY- Sub/Amp Install With A Stock Radio - Hyundai Forums : Hyundai Forum
All you might wanna do after that point is research some better than stock tweeters to swap out in the front.
After that you would have an awesome system with the stock head-unit and you keep all your steering wheel options and clean appearance.
Any questions or comments u can message me.
Hope that helps. repko