but then we'd have to pay attention to how people actually voted!
No, it would actually come down to, "But then we'd need faith in the accuracy of our vote counters"
If you know that every vote is being counting properly the first time then election via popular vote is the perfect solution. But until that day comes I'd like to hear a better method of quick and (mostly) accurate results. Time is the deterrent this is senariro. The american people want to know who their next president is within hours of voting. Perferably without year long recounts. Would you like to multiply the Florida recount across the entire country? Because you just know what someone, somewhere would lodge a complaint and ask for a complete recount.
New systems generate new problems.