Of course it would be good if FIAT and crypto could get along but those worlds are opposed in nature. The reason they are interested in crypto could be varied but it is likely to free themselves from the fiat system as it currently stands.
That would be the ideal situation but fiat and cryptocurrency would be compatible with each other. Currently, some countries view bitcoin as an alternative payment to their services/products and you also have the option to convert your bitcoins to fiat. Depending upon the government situation, bitcoin and fiat may be possible utilized but it will create a conundrum between the two forces as they may conflict and cause further conflicts in the transaction aspect.
I believe that once bitcoin has been legalized, it will eventually become a currency that will be existing electronically- which will ultimately eliminate the purpose of peer-to-peer transactions.