You log into GOG then enter your Steam login, and the games are added to your library.
OR (hear me out) you could log into your Steam account and just play the games. Radical idea, I know, but give it a whirl. Super easy and doesn't require multiple digital clients or logging into one client through another client.
Steam adds plenty of hassles such as offline mode failures, accounts wrongfully locked/banned, authentication issues, startup bloat to launching games, forced auto-patching, a bloated client that's forced to constantly run, corrupted game caches needing re-verification, and the fact that after a Windows reinstall you have no access to your games until you install it.
I've been using Steam for over a decade and the only issue I've had is my games randomly not wanting to play in offline mode. I was without internet for about 10 months and I was able to play Borderlands 2 every single day until randomly it said it was queued for an update, so I had to take my PC to my buddy's house just to get the "update" (sync issue) then go back offline. That happened maybe two or three times in over ten years. The rest I have no idea what you're talking about since I haven't encountered any of those issues, nor have I even heard of them before. I've heard of people complaining about their account being VAC banned but there was always proof provided for the ban or an account being locked because "my brother got on my account and (insert sorry ass excuse here)".
Also, saying that you don't have access to your games after a fresh install is kinda redundant, don't you think? You don't have access to anything after a fresh install until you actually put them back on your PC. That's like saying you don't have access to photoshop until you reinstall it after a fresh OS install. Owning physical copies of the games are no different because you won't have access to them until you actually install the game. For Steam, you install Steam and your entire library is there waiting for you. I'm sure it works the same way with GoG, Origin, Windows Store, Uplay, etc.
If GOG shuts down it doesn't matter given you can backup standalone DRM-free installers for all your games anywhere you please. No lynch mob needed.
Such a thing is called ownership, an apparently alien concept to many. Some people feel ownership and consumer rights don't matter as long as Steam gives them trading cards. "My precious shiny", and all that.
You really need to loosen those straps on that tin-foil hat of yours. Not good for your brain, plus it makes your head sweaty.
With Steam, there's always the possibility (regardless of how small) that the client could one day just not exist. We all know that, but the likelyhood of that happening is so small it's not even worth worrying about and it's definitely not a big enough issue for anyone to run out and buy hundreds of blank DVDs to burn backup installers. Chances are, those discs will corrupt long before the digital distributors go down, especially considering how many people are switching over completely from physical to digital. Hell, there's apps now that let people do their grocery shopping online - it even keeps track of your purchases so you can just go back in and add quantities to your regular stuff (milk, eggs, etc.) the next time you do your shopping. If people are forgoing actually going to grocery stores in lieu of shopping online, it's a safe bet that gamers (who are stereotypically less inclined to go out in the real world and interact with people /joke) are not going to suddenly prefer physical copies.
That being said, I do understand where you're coming from that it's better to be safe than sorry. It would be a good idea for me to build a bomb shelter under my basement just in case, but the chances of actually needed it are about as slim as Steam going offline permanently.