Yes. You could use this to effectively chain the ampunt of status moves you use in one turn, but you'd also lose several Pokemon as a result.
Here's an example

Let me start by saying I'm not aure this is balanced.
Glimmora leads and uses Memento or Spikes/TSpikes, whatever you feel is safe. After death or memento, Gardevoir switches in.
The switchin Gardevoir will now use Spikes, Toxic Spikes, Stealth Rock, and then KO itself with Memento given to it by Glimmora. Switchin Gallade.
Gallade uses Disable, Encore, Wish, and then Memento given to it by Gardevoir. Switch in Dragonite.
Dragonite uses Mean Look, Swords Dance, Agility, and then Imprison. Coupled with Dragonite owning Haze, this whole setup, while it sacks three of your guys, makes Dragonite really, really hard to stop. There are of course plenty of ways around this (Heart Swap over Haze, GaG/Magic Bounce, Ghost type/tera ghost, having your own 3-mon sack setup, etc.) but I'm not sure if that should still make this allowed in principle. Getting to use 12 status moves in a turn sounds fun though.
It's gor a few differences!
- Use of the extra status moves is a lot more limited, given it only activated when a Pokemon faints.
- You can use multiple status moves in a row.
Parting Shot, Teleport, Baton Pass (not sure if needs a ban since you could just choose the mon you want to have the stats directly, maybe Numel could be looked at though), and Chilly Reception would all activate once.

has Parting Shot and faints whule active. The switchin

would be able to use Parting Shot, but the switchin after that wouldn't use Parting Shot since

has not fainted and does not actually own Parting Shot.
Having thought about it a bit more, I think Roar, Whirlwind, Shed Tail, and Substitute are all potenrially a bit too strong as well. Roar & Whirlwind are strong setup tools, allowing you to potentially force out a threat and immediately setup upon one of your mons fainting. They also dismantle setups too by simply yeeting them. Haze does this too, but I think the fact that it doesn't switch mons out keeps it a but more balanced.
Substitute & Shed Tail are pretty similar, in that they can make setup a bit easy and also make opponents who wanna make use of this meta less able to do so (as sub will block several status moves).