I'm not looking for extra speed. I just really like the way the SRAM double tap shifters work. It's, to me, much simpler than the Shimano stuff (which is great). I don't feel line I need to swap the front since I shut so much less up there. And it was a little bit curiosity about the differences between the systems and how important it is to match the front and rear. It sounds like the match of the front and rear is not that important other than to ensure that the chain will work with both. I think the one other considering would be that the brakes (disc in my case) will also work with the brifters.

That’s entirely wrong. The shape of the front deraileur is very important to trim the chain so it doesn’t rub as you shift gears in the rear. In addition some manufacturers use different shaped derailleur cages within the same model to mesh with different size chainsets. Double and triple chainsets often require their corresponding derailleurs and shifters and there are short, medium and long cage rear derailleurs, which also differ with push or pull cable systems and traditional or direct mount. Different brake levers have different amount of cable movement so a disk brake “brifter” lever might pull too much or not enough for a caliper brake.
So the whole system is designed and spec to work together and often times, a large manufacturer like trek or specialized will have shimano build a component to their spec but not be made available to the aftermarket. We often had people who purchased components in an attempt to upgrade, bring their bike in because it ran worse than with the old stuff. That was 20 years ago. The situation is even more difficult now. Then you add in new components to an already worn system and the trouble is compounded.
At the very least, ask your local bike shop geek if what you are buying will work with what you have.