I know of only one way that this can happen in windows, and has nothing to do with Steam other than their not-so-great decision to install all content into Program Files.
If you are on Windows 7 or Windows Vista, and you have UAC set to ON, then change it later to OFF, this can happen.
UAC does thing evil thing where if a program tries to write to a "protected" area like Program Files, EVEN IF YOU ARE ADMINISTRATOR, when you get a UAC prompt and give permission, it actually writes any files that would normally go into that directory into a "virtualized" directory somewhere else.
If you turn off UAC, that will no longer happen, and as an awesome bonus, it will no longer know about the files that it virtualized.
So in your case, if you had UAC on, installed a ton of games, then later got sick of UAC and turned it off, this would happen as you described.
See the second paragraph in "Features" in the article on User Account Control.
If you have done this and turn on UAC again, your content will come back, but be a TOTAL mess because if you've downloaded more stuff with UAC off, then that will be invisible when you turn UAC on, and vice versa.
this drove me a little crazy once before I realized what was happening.
i'm curious to know if this is your issue.
Games that require Steam cannot be legally played without Steam. Steam is essentially a form of digital rights management in the context of these games.
Steam is approximately 100MB to download, if you have a slow internet connection you're probably best off leaving it overnight to download as a workaround. If you really can't do this then there aren't many options left.
Once Steam and your game are both installed, you can put Steam into "offline" mode, which will allow you to play your games without being connected to the internet.
Best Answer
Your comments are different than your question, but I'll try to address both:
You cannot by default prevent Steam from closing games in situations where the Steam client itself closes after being used to open them. To help with this you should close the game and give the client a moment to finish syncing with the Steam Cloud before letting it restart/close, because even if the game doesn't use Steam Cloud the client is still going to update some config files. If it's not allowed to write & sync those files it may cause a crash, meaning the Steam client will run its updater on the next startup to make sure it's not broken and some configuration data may be lost.
The reason you can't leave them running but close Steam is partially because they are attached to it through SteamWorks to send data to the client, and another part may be because of how launching an external process is handled on operating systems (usually attached as a child to the parent that opens it, and so they die if it closes).
It may be possible to modify the client somehow to add functionality to "disown" a game's process so it's detached after launching, though that would almost definitely break SteamWorks integration. Where possible it would be easier to just launch the game through the executable directly instead as this would be some extremely shady and likely also difficult programming.
To find the executable of a given game you have installed, right click it in your Library and go to Properties -> Local Files and press "Browse Local Files". This may have changed slightly with the UI updates over the last couple years, but should be similar still.
There is a forum post on GOG.com from 2015 about DRM-free games on Steam that's still actively commented with additions, and it points to https://steam.fandom.com/wiki/List_of_DRM-free_games, which is still updated but may not have everything the forum post comments has.