The in-game server browser has been broken since around ~2015

People wanting to join a coop game (except the host) must use the modded server browser available on moddb or github
https://www.moddb.com/games/cry-of-fear/addons/server-browser-for-cry-of-fear
All credits for this mod go to Bernt Andreas Eide/Pretador and their collaborators

Download the .rar file and extract anywhere (BUT NOT IN THE CRY OF FEAR STEAM FOLDER, IT WILL BREAK THINGS)

In the CofServerBrowser.5 folder, double-click CofServerBrowser.exe to see the available servers
(Also read the readme.txt for extra information from the author of the mod)

Double-click the server you want to join.
If Cry of Fear game wasn't already started, you might have to double-click a second time on the server you want to join.

In-game voice communication seems broken too, no fix for that

The host must forward/redirect their router's port number 27015 in the administration tools of their router.
Your interfaces may vary during this step depending on your ISP and/or your hardware.
You're looking for menus called NAT/PAT/TCP/UDP. Given the opportunity, forward both TCP and UDP for the port 27015

I suggest reading https://www.noip.com/support/knowledgebase/general-port-forwarding-guide
The readme.txt file in the modded server browser of the previous section also has useful info about port forwarding.

When you're done, reboot your router (you will lose access to Internet for a few minutes)

The host can then create a Cry of Fear coop game as usual using the in-game menus.
If the menus are cropped out of existence, you might have to decrease the resolution and enable Windowed mode in the Cry of Fear launcher.

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Other users with the modded server browser can now see and join the host's game