I was trying to find a way for my Steam games to show up in GOG like you can do with the "non-steam game" option on Steam. Does anyone know if there is a way to do this? The only option I found was connecting games, but none of mine were eligible.

No, to my knowledge no such feature is available in the GOG Galaxy client so far, nor the website. GOG Connect is for getting GOG copies of select games you own on Steam, so that's a very different thing also.

Yeah, basically it's just running other games through the Steam overlay so that you can access the chat, screenshot functionality and stuff. GOG will probably add something similar for Galaxy at some point.

It's one of the few reasons I still use it after transitioning to GOG. I wouldn't mind if Galaxy adopted a couple of its other remaining features (please no trading cards) and allowed us to plug our other games into the client but I'd rather not encourage them. I prefer it to remain a tool and not a necessity.

These threads might interest you :

 ▪  Adding external games to Galaxy
 ▪ Non-GOG game adder for Galaxy - download

that does not work on mac for example, also it's not convenient.

GOG only consists of game installers and a client. The game installers just install a game that is started, no way to show any other game there. The only way would be the client and afaik there is an option to add something there, but I may be mistaken. If it doesn't exist, it might be on the way.

However, many Steam games would still need the Steam client to start. So if I understand you correctly you want the GOG client to display and start Steam games by starting the Steam client and telling him to start some Steam game? The other way around is much easier, Steam just needs to find the installation of the GOG game and start it. So I don't know if this should be a goal.

Also, do you want anything else besides the games appearing in a list and being able to be started? What exactly does Steam with adding a "non-steam game"? If it's just about the list, maybe another third party service exists which can show you all your installed games and let you start them with a single click.