As a few others said, the progression is designed for continuous play. Maybe viable if they did something like Half Life (The crappy Source Engine port, not the original), cause I think in that the game would give you a load out of most of the weapons you were supposed to find before the chapter you select.
This feels like asking for a lot, especially given that they need to set up netcode, and everything else for a 2004 game that didnt have such multiplayer features initially. I know quake got that when it got updated, but it already had the multiplayer functionality in the first place. Chances are though, you can find a decent fan made mod adding at least coop in, given Doom 3's engine is open source.
EDIT: also, this...
Already done in the Re releases for Doom 1 & 2.
Multiplayer has been in Classic Doom for years? Even in the earliest days of it, Ultimate Doom had support for LAN duels IIRC.. And Doom 2 definitely had the multiplayer support, and many clients for it, like DWANGO, have been made for it back then.
And there are also multiplayer oriented ports like Zdaemon, Odamex, and Zandronum for today. You also need a server browser to connect to servers more easily. Zandronum should come with one prepackaged.
What about Discord or Skype? Consoles like the PS4 and Xbox also have options where you can mix party audio as well as game audio, as well as choose to play in game chat or party.
There is no way in hell they're gonna see this and take your request to heart.