There was a situation when Fisk could easily remove his mask and reveal the true identity of the Man in the black mask. Why didn't he do that?
Do you think Wilson Fisk knows the face of everyone in Hell's Kitchen?
He doesn't. So let's say he pulls off the mask, and reveals this face.
Odds are, this person is not a public figure, nor is he in any way known to Fisk or his associates. Daredevil fights crime in general, and while he of course focuses on Fisk too, that doesn't mean that he's personally connected to him.
And even if he knew who the person was (after an extensive search), then what? This guy is still going to keep on doing what he does, regardless of whether you know his face or not. If anything, he'll be more incentivized to shut you up to keep his own identity a secret.
You even answered the question himself: Fisk wanted him to die then and there. There's little point to knowing the face of a man you don't know.
Not knowing the man's face would even help Fisk with plausible deniability, if the police one day show up on his doorstep with this picture and asking him is he has ever seen this person before.
And then there's also the storytelling perspective, where revealing the identity of a superhero is generally avoided unless it's relevant to the plot. Since it wasn't relevant to the plot, there was no point in making the viewer think that Fisk's knowledge of Daredevil's identity will develop the plot in a different way. It distracts from the actual plot.
- Fisk was having him killed then and there. Knowing his face served no purpose.
- There are significant odds that Fisk wouldn't know him by his face anyway. Due to Daredevil's general crime-fighting agenda, there's no reason for Fisk to think that Daredevil is personally connected to him.
- Who says that Daredevil has a personal life when he's not fighting crime? It's just as likely that he lives off the grid; thus making it impossible to find him based on his face or name. Frank Castle is widely known by the public (due to the events in DD), yet he still keeps on doing his thing. Frank is nigh untouchable because he has no personal life that Fisk can use against him. Why would Fisk expect Daredevil to be any different from the Punisher?
- If Daredevil somehow escapes, he's not going to stop just because his identity is known by Fisk. If anything, revealing his identity would make Fisk a personal problem to Daredevil, which he currently isn't.
- Since it wouldn't have been relevant for the plot, a pointless face reveal was omitted. This is a correct application of Chekhov's gun.