I don't know the details, how DAT became SN Brussels. As far as I remember, it was more or less the same, as with Crossair, mainly an airline based on the former regional airline under SN (DAT -> SN Brussel Airlines, Crossair -> Swiss International Air Lines).
Let's have a closer look at SWISS. After the grounding of Swissair, the SWISS government injected a few mio CHF (don't remember the number right now).
Then, after looking at all the possibilities, it was clear, that Swissair wouldn't survive, but only with some bn of CHF. The other possibility was, to go under "Nachlassstundung", which can be compared somehow with the chapter 11 thing in USA. UBS and Credit Suisse, the 2 iggest banks in Switzerland bought Crossair from SAirGroup. The SWISS government and private companies invest around 2 Bn Swiss Francs in this project, which had as goal to build up an intercontinental airline based on Crossair. The Swiss government also reserved about a bn CHF to guarantee the Swissair operations until 31. March 2002. Since then Swissair doesn't operate anymore.
SWISS can't use the name Swissair due to legal consequences. FIrst of all, the SAirGroup is still the owner of this brand and they want money for that brand. SWISS on the other hand didn't want to pay that much for that name. Another reason for not to buy this name are legal problems associated with the bankrupcy of Swissair. That case, SWISS would have to fear to be forced the depts of Swissair. Even so, there will be legl problems, as seen in France for example.
Hope that helps a little bit and this post was WITHOUT ANY BASHING in any way.
dominguez(dash)online(dot)ch ... Pushing the limits of my equipment