I don't know for sure, but, as a hypothetical example, in this case, MTV could have requested that IA take down the archives from that time period.
(That is, if MTV can prove they owned the domain during the archive period [probably pretty easy]), and if they still hold the copyright to that work [probably], then they could essentially issue a DMCA takedown request [by email], with which IA would most likely comply [there are some HN and IA forum posts about this I think].)
> just because the parent/ source website was shut down
The "just because" part is not a well-founded assumption AFAICT. For all we know, there could have been a request like above.
> why wayback machine and the other archive services aren’t filling in this gap/working in this case?
I see no evidence that IA isn't working in this case, if working is defined as including having a process for handling valid takedown requests that must be acted on according to the rules they follow.
I hope that helps. Again I don't know if this is what happened, it's all hypothetical.
The analogy that comes to mine in regards to MTV taking their full site off-line, would be the equivalent of a newspaper shutting down and destroying or burning all of the extra printed copies of their paper/articles they may have in their own archives instead of donating them to a local library.
So not all, only "older" articles are not available. Wayback Machine only works after someone submits the site for archival; the old site might have used something that made it hard to archive content; and I expect MTV pre-dates the Internet Archive. Any of those factors might result in WM missing stuff. It's not magic.