This is a great subject, because it is fairly easy to see the impact that Bartok and Kodaly's work has on what is considered Hungarian music(and folk and traditional music in general) by simply checking out the record bins to see way that   recording of traditional music and performers who try to recreate traditional sounds and repertoires(based a lot on the foundations of both Kodaly and Bartok) have come into the mainstream.

First thing to understand is that folks are getting the gypsy music part of it wrong--we are not talking about the traditional/folk music of Hungarian Gypsies, we are talking about the entertainment music that Gypsy performers played for popular audiences(lots of non-gypsies played it too)--
We are all familar with this type of music, because we've seen and heard it so much in old movies, on TV, and on the stage--it is the "strolling violinist" sort of stuff that is characterized by theatrical emotions, frenetic solos, abrupt changes in tempo, and especially the "locomotive effect" which starts out with a heavy, plodding tempo, and increases gradually and overdramatically, to a breakneck speed and thundering crescendo. It was/is music more to be seen than heard, and is a sure applause getter--

Classical composers and performers of the 19th century including such folks as Liszt and Pagannini, drew heavily on the musicial tricks that Gypsy performers used to get tips, and exploited/created/perpetuated the cultural stereotypes of mysterious, passionate, dark, sinister, gypsies to market their materials--it became standard practice for performers to claim a bit of gypsy blood in their ancestry to lend credence to their performances--and, --to a certain degree, it still is--

Some of the music that was created this way was great, without question, but a lot of it was really tedious, and even the great stuff wore thin after a while. (excluding Bugs Bunny Cartoons, when was the last time you heard or wanted to hear the Hungarian Rhapsody?)

Anyway, the idea that there was real traditional music out there, including really instrumental performance traditions, and real culture to go along with it was very appealing(and still is)--especially for young composers who always need to find material to work with that distinguishes them from the previous generation of composers--

For Bartok, it strikes me that discovering tradtional music was like WC Handy's discovery of the blues--it was new and familar at the same time, and, because he could claim it as his roots, instead of being just another classically trained composer, it gave him a heritage of his own-

As far as the idea of creating at national music,It was (and is)a very useful way for groups of people who are trying to keep things together while empires are falling and being toppled around them to find an identity that is not tied to their rulers or opressors. Furthermore, finding that identity in folkways, like dress, food, music, and dances seems very appealing, because it is so fundamental.

There are a couple of problems though-- folk/traditional music is a local phenomenon--folks down the road, across the river, and over the mountain often play and like different stuff, so a "national music" would be artificial in a lot of ways. If the work of Kodaly and Bartok proved anything, it proved that performance traditions and repertoire could be entirely different from one village to another, and that a "National" music could not naturally exist--

Probably more important, defining a culture by shared traditions means excluding those that don't share the traditions. Also, gathering people together based on a shared past implies a shared destiny, and strongly implies that that destiny is not shared by those who don't share the traditions--no coincidence that the Nazis relied heavily on folk icons--

As a long-time Balkan/Middle Eastern music junkie, I have seen many instances where folk music traditions, real and imagined, have been used as a rallying tool with disturbing racial overtones, and have seen those overtones played out in the headlines and I continue to see them.