Business is a perfectly acceptable career for both children and grandchildren of the monarch in the Netherlands, all styled Princes. I don't know why Denmark would be any different in that regard.
I am not sure if you have seen this post by Muhler, but I think it explains it well.
To the general public who are not that well versed in titles, there is no real distinction between HH and HRH. - Nikolai is Prince ergo he is a full royal. That also applies to segments of the press, especially when they want to...
Nikolai can very much be compared to Princess Elisabeth I think.
She's never married or had any children, so that "issue" ends with her. And she had a career in the Foreign Ministry, sometimes being deployed abroad. That's an acceptable and safe career for an "inactive" royal. Not controversial and out of sight and out of mind.
And if need be I suppose all Joachim's children can be parked away safely like that. Beforehand younger royals had a career in the military or like Elisabeth became a civil servant or even went into the church or in some cases had a manor somewhere. - Pretty much the traditional career choices for younger siblings of the upper class.
But that does put limitations on their choice of careers.
Nothing that is remotely political. Nothing that brings him into conflict with Danish interests or with segments of the population is acceptable, so that may rule out him being a top executive in a private business. And so on.
The problem is also that the DRF has to neutral in almost every aspect. That also includes inactive members. Otherwise there is a very good chance of that detracting from the monarchy itself. - That sentiment is admittedly sometimes bordering on being hysterical here in DK, but it's there.
It's easier and simpler to renounce the title and just be a count. For Nikolai himself it won't make that big a difference in relation to the DRF. He'll still be invited for family events and such, but not standing with his cousins. I think he'll survive that. It's not like he's being kicked out of a palace or anything like that.
So if Nikolai and his siblings wish to retain their titles as royals and have a career of their own it's better if they settle and work outside DK.
That I think applies to the Nordic monarchies. They have different traditions in the Benelux countries, and yet, even there there have been controversies and scandals when secondary royals have messed things up.
So again: If Nikolai is inactive, it would IMO be simpler for him to officially opt out of the royal roadshow.
We debated something similar in regards to Madeleine when she married. I think she would get much less criticism and be much more free (and so would her husband BTW) had she renounced her royal title when she married.
Now she is criticized for not being active enough, for not being in Sweden enough and so on...
I am not sure the Netherlands is the best example to cite regarding business careers for princes. There has been abundant backlash over Prince Bernhard of Orange-Nassau's real estate career and lesser controversies over perceived or actual preferential treatment of other princes in their careers. Of course, it is clear the Dutch public still finds this much preferable to supporting all the princes with taxpayer money, but it is not a flawless model, and I assume similar concerns contributed to the Dutch cadet branches losing their princely titles in the next generation, beginning with Prince Constantijn's children.