I don’t think the New Testament implies that the Apostles were known as Apostles because of their garments. They were known by their zeal for the Gospel. If a priest couldn’t be recognizes as a priest if he weren’t wearing a collar, that is a problem. If he’s not wearing a collar because he doesn’t want the constant expectation that he will act as if he has made the radical commitment to the Gospel that he has, that is a problem. If anyone who knows him could not mistake him as anything but “if that guy isn’t a priest, he ought to be,” then I don’t see how failing to wear a collar is a major problem. My opinion is that the priest ought to comply with what their superior requires, though, and I’m not their superior. It is the religious orders decide the dress code for their own communities, and bishops who make these decisions for diocesan priests.
As far as accepting being referred to as Father, it depends on whether they are taking “all no one on earth your father” (Matt. 23:9) to an extreme or whether they are shrinking from being identified with what is, yes, a radical call. That is a bit more problematic, as it implies they may think priests who do accept that honorific are being pompous, which is not true.