No, it does make sense. Because the FAA license you get on the basis of the EASA (called a piggyback) is only valid if the EASA one is - and has all the limitations of that. For example, you can't fly at night on a FAA license issued on the basis of a EASA license unless you have an EASA night rating. But if you do stand alone FAA license, then that's part of the PPL. Also, any ratings etc are only valid if they're on the EASA. Lose your Euro medical? Can't fly on the FAA one either, etc, etc.
Best option is to have two separate full blown licenses, that way you are not limited to the limitations of one. It could even be so simple as you develop a disease that the EASA countries won't issue a medical for, whereas the FAA will. There are numerous such examples like HIV, diabetes, anti-depressants etc.
I did my full blown FAA license even though I had a JAR one myself as I didn't want to build any ratings on an FAA license that was a piggyback. Like an IR would then become invalid should my EASA expire etc. It costs a little more and takes a bit more time, but it's worth it. And especially if you're doing the integrated way you're doing, then that's the best way as most of the basic flying manoeuvres are straight crossovers.