Other mystery companies have already offered GOG money to merge, they even confirmed it in an old series of live Q&A videos. They've denied all of them because they wanted to grow it themselves.
What are you asking specifically when asking this question? Do you want GOG to have a more complete client? Add DRM? Have a larger community? Have more general attraction and praise? Or are you specifically talking about the games?
Or maybe all of the above?
Steam was the first to the scene all the way back in 2003 or 2004. I remember the ugly puke green client like it was yesterday. It was absolute fucking garbage and I remember a LOT of people being upset. Especially in the many old Half-Life forums I used to venture. Many people swore they would never use such a garbage client or support Valve's idea. Some even predicted it would be the death of Valve and such foolish decisions would surely sink them.
Connection issues, download issues, client glitches and bugs, it was a very rough start. Yet, here we are in 2016 and Steam is #1 in the market. The same people who swore they'd never use it have hundreds of games on it, and Valve is far from sinking now. Even Microsoft is jealous of Valve and realizing their neglect towards the PC market is hurting them a bit.
To say they haven't earned that success is a bit narrow-minded imo. They've clearly worked to gain customer trust and entice people to deal with their DRM by making their client very community and customer focused (for the most part). That's why people always list the social features when asked why they like Steam. Streaming, FPS counter, screenshots, achievements, profiles, friends lists, chat, music player, in-game markets as seen in TF2 or CSGO. These things matter to people and makes them feel more "connected" to the client and as a result the loyalty forms. Add to that the fact that Valve's head is a pretty nice and jolly guy that the internet seems to love and as a result their image forms. Jesus with Gabe's face photoshopped into it.
Did they save PC gaming? Not likely, if not them someone else would have come along, but they were the first, they caught everyone off guard, everyone doubted them, and now they are #1 through their own work.
It was such a success that now other major companies are trying to copy their business model. EA has Origin, Ubisoft has UPlay, Microsoft has their.... thing.... and now even GOG is trying to emulate them. What other purpose would they spend the time, money and resources on developing a client? They want those AAA big games. In many ways I'd argue in order for GOG to directly compete with Steam, they'd have to emulate the things Steam does and do it better, and right now as we speak they are already trying to do just that, and just like Valve the start to their client is seeing a very rough start. The only major thing that separates it from Steam is the fact it's "optional". The only question is, is that going to be enough to attract publishers to bring games, and is that going to attract customers? Or are most people going to question the value of rebuying their Steam library here if GOG is "pretty much the same minus DRM". Because I can tell you that most people honestly don't give a shit, or are willing to accept systems like Steam because it gives them a reason to deal with it. "CSGO is there, I like the social features, all my friends are on Steam" whatever it may be.
So OP, I'd answer this by saying GOG is already doing just that. The thing is, a lot of people are going to be split on that fact, because when GOG started, it didn't even want to be anywhere near Steam. It simply wanted to be a store that sold games no longer sold or games that no longer worked on newer hardware, or both. That's it. Unfortunately, at some point, they hit a wall and realized that focusing only on older games would be the death of them. So they re-branded, no longer Good Old Games, but just GOG. Then they started accepting indies and newer games, and movies, and started working on a client, now you can link your account with Steam, a direct nod to Steam and a direct move against Steam, things started changing and fast. Do I think they'll ever be bigger than Steam? No. Honestly I just don't see it for the simple fact that most of the biggest PC games are thanks to the MP communities, and I think this is the key thing GOG cannot emulate from Steam. Those are the games that bring in the big dollars. Steam has their system that makes it easy for devs to manage that, and I'd be willing to bet most people won't rebuy a game they've logged hundred of hours into MP on Steam just because it was DRM free now, and with the rest of their library already on Steam, are not going to be given enough of a reason to just up and move over.
GOG is probably the best alternative to Steam right now, but honestly that doesn't say much at this point. Sure, there are lots of great games on GOG, especially if you love all genres or older games, but Steam has more. A lot more. Old games, new games, indies, applications, movies, it just has more. Some of these games which will never touch GOG for technical reasons, or publisher reasons, and I get the feeling GOG doesn't want to be considered an alternative, they want to be a direct competitor. I highly doubt they'll ditch their DRM-Free rule, because that's the big difference from Steam and others, but that's why they were willing to ditch most of their other pillars.
Of course I could be wrong, a repeat of history perhaps? "GOG will never be like Steam! What fools!" only for 5 more years to pass and they overpass them? Only time will tell. I'm one the side that, GOG will be around sure, but they'll always be #2, and that's not really a bad thing. I personally use both, because I want to play games only found on Steam like Company of Heroes 2 or Insurgency.