A lot of people don't realize the political and economic benefits of having a monarchy. The Danish Queen is followed around by an entourage of ministers and businessmen when she goes on official visits around the world. This opens up for trade relationships that would otherwise not be possible or simply wouldn't take place. In addition to this, having a monarchy which is separated from the political and legislating body, allows for Denmark to remain 'friendly' with countries we're otherwise in conflict with.
Yes, the Denmark royal grandchildren will fight for a paycheck. Why? Because it is the human condition to want money for doing absolutely nothing. How many people in the United States alone get all worked up when the remotest of possibilities are posited to cut back or remove government aid? Aforementioned takers need to expend as much energy protecting their free lunch towards getting jobs.
To me, it makes perfect sense to stop paying allowances to the royal grandchildren of Denmark. The only reason they received an allowance was because they are of royalty, not because they did some form of work to earn it. I doubt they will try to fight to get it back since this idea seems to have a lot of support.
No they will not fight for a handout. They would not know how to fight to even support themselves. When you have lived a life of thankless handouts, how would you even know where to begin to fight for your (un)deserved money. Maybe they will have to get real jobs now.