We'll have to agree to disagree and that's not what you originally said, you claimed you couldn't run aftermarket speakers with the power that the stock radio puts out. You said "Sadly, there's not enough power to run anything but stock speakers in your situation."
Absolutely it can be done and with significantly better sound than it had with the factory speakers. I spent years in the very high-end audio industry doing up to and including theaters and stadiums and full-well know the requirements of getting great sound. YES adding an aftermarket high-power amp will help when the Jeep's top is down, with a soft top, etc. but you can still get a significant improvement with nothing more than good quality aftermarket speakers with the STOCK radio or aftermarket head unit providing the aftermarket speakers have a 90 dB or higher Sensitivity rating.
Before adding the aux amplifier to my last TJ I installed Polk DB speakers with the factory head unit and the improvement in sound the Polks made was very significant.
Your idea of significant improvement is probably different than mine because a stock head unit running Polks would not be what I'd considered a significant improvement. In fact, I'd be pretty bummed as soon as I drove it down the road at a reasonable speed.
The jeep's environment, envelope, power availability, etc. are very different from a stadium or a theatre. We are discussing car audio decks and four jeep speakers where tires, wind, and road noise play a part in the listener's experience.
With that said, I should not have said it can't be done (obviously it can be done). I should have said that it can be done but will almost certainly have practical limitations that can easily be improved upon (by doing more than what you suggested).
I prefer to build anything I do to a standard that holds up to most if not all of my practical uses. I drive up to highway speeds with a soft top up or down w/ 35s, and I'm still able to enjoy my music. I didn't have to do anything exotic to achieve this level of performance. And I don't have to sit in idle, drive slow, or let off the throttle to be happy with my tunes. No mind tricks are necessary for me because I don't have "kinda good I guess" results.
You don't really explain your reasoning for your significant-improvement claim or where there might be limitations to the proposed significant improvement. It would be super useful if you told us why you believe there is a significant improvement along with environments where you found that opinion to excel or be limited. Providing your reasoning is very important. I'd like to see more applicable details for your reasoning rather than you pivoting into your past work on stadiums and theatres, which involve very different variables and constraints.