I think a VHS tape was used in that scene in order to make it easier for the audience to quickly identify the media as a one-off custom recording and not a mass market DVD release. It fits the story’s time frame better also.
I think the first half of your comment may have nailed the reason. I got to disagree with the second half though. It is understood that the story starts in 2006, and by then VHS had pretty much been left by the wayside.
There also may be a yet to disclosed backstory reason that Gillgan used that effect. With BB/BCS, you just never know. In the vein of tarantula in the jar, or pink teddy bear eye in the pool kind of stuff.