Of course, if one employee being away would cause a company to fall apart, us Dutch would conclude management is completely incompetent, and tell them that.
A 3 week vacation is pretty normal, here. But we do plan those ahead. That means you might not be able to take it on the specific dates you have in mind. But not that you won't be able to take it!
Actually, our cats often join us when walking our dogs. We have two dogs, and when we walk them leashed, at night, one to three of our (six) cats come along and walk around us. They dart out in advance up to ten meters, using available cover (cars) to hide, and often laag behind in those places to ensure it is clear that they are not being walked but are simply following the same route of their own accord.
Not sure how this happened, so I can't help you with training advice. But maybe it's just our regular schedule of walking that does it?