I've never tried the EC, but I can say that using underwear at home and diapers on the run are awesome for potty training. My first daughter potty trained this way at 16 months. (don't get too excited, she was the only one out of 4 who PT before 2 yrs.) I actually remember being able to bring her out and about with only undies on at 16 months. Everyone thought I was sooo brave. But I simply didn't see the need to put a diaper on my daughter who hardly ever had accidents :)
I actually have a rule now with my 30 month old who refuses to potty at home, but fusses to go when we are in public. No public potties until you can use the potty at home!
Needless to say, I'm cautiously optimistic.
I wouldn't worry too much about your carpets. Our son has been in undies since 16 months (now 20 months old). We find that pee am poop accidents don't end up on the floor, couch or carpet unless he's naked. If he's is wearing undies and pants, any accident is pretty much contained in his clothes. It means plenty of costume changes, but carrying extra pants and undies is standard for toddlers no matter how they potty train. Also, pee and poop wash away with soap and water, and don't stain.
A useful trick we learned for pee leaks: if it's somewhere you can't wash right away, like a carpet or your bed in the middle of the night, a spritz from a spray bottle filled with 50% vinegar, 50% water will make any pee smell disappear for as long as you need until you can wash it.
I'd say, don't be scared of the mess. There are messes, but not as often as you might think, and they're no harder to deal with than a diaper change is!