The following interview (a longer version of the video you linked to) explains the reason for both the dual voices (which was actually Kevin Conroy's idea and had only been done before once by Michael Keaton) and that he ultimately had to tone the Bruce Wayne voice down. It doesn't mention him ditching the voice completely but it seems likely that this decision came even later from the creators due to the points he brings up in the interview.
Transcript starting around 4:26 but I recommend listening to the whole thing.
When the first group of shows came back they realized it was much darker than they had anticipated, and they loved that, but that the voice I had been using for Bruce Wayne, which was sort of like the voice I'm using right now, was very, you know, kind of charming guy, you know, just me, um wa a little too light. It just seemed too funny. I was actually using a lot of humor, I was doing things, a lot of sarcasm, um, he had a real sarcastic edge to him. Which I thought was great to contrast the Batman character, um, and it was, but it didn't work in this noir dark world that they created. So I actually had to go back and rerecord the first like six episodes, uh, just the Bruce Wayne voice, just to tone him down, to make him closer to the Batman voice, because they liked the distinction idea but it just had to be very subtle.