Hey. It's been a long time since I messed with Wine and Crysis. That was before the Community Edition, before the launcher and before our fake Gamespy server. So bear with me.
The launcher is a necessary evil. It handles signing into "Gamespy" (or rather our replacement dummy) for Crysis, among other things. It's not intended to be able to launch Crysis independently, and it won't function properly without the launcher magic, as you've noticed. Your best bet is to get the launcher running.
The delay after clicking Internet Game sounds like DNS. We had to hardcode some domains in the game's binary (such as the defunct server browser chat) to invalid domains. If they actually resolve (usually due to DNS hijacking), the game will try to connect to something and only time out after a few minutes. Monitor DNS traffic while clicking the button. Also see here.
But unless you're using the launcher, you will still be in a non-logged-in state, or in a state where game servers will reject you, or server admins will ban you on sight.
@fire-hound might have some more recent experiences with Wine: https://forum.mechlivinglegends.net/d/161-playing-or-trying-mw-ll-on-linux-via-wine
Fun fact: The Funbocks servers have been running on Wine since the very beginning! They were a very different beast to tame compared to the client.