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It sounds like what you want is:

>There is a program installed in the regular OS on my hard drive, but I am booted into a live OS. How do I run that program?

The common way to do this is to use [chroot](https://man.archlinux.org/man/chroot.1).

Linux looks for programs under certain standard paths like `/usr/bin`. If you're in a live OS, your `/usr/bin` will be on a virtual disk (which uses your RAM) and will contain programs that came with the live USB. The normal OS's files are in the `/usr/bin` of the drive, which is not active in the live OS.

First you have to make sure the normal OS drive/partition is mounted. In some GUI distro live USBs, you will see your drives in the file manager, and when you click they will automount at something like `/media/my_drive`. Others expect you to manually run [mount](https://man.archlinux.org/man/mount.8), which traditionally mounts at `/mnt`. 

Let's say you mounted your main OS partition at `/mnt`, so that you see `/mnt/usr/games/steam`. You can't run `steam` because Linux will look for it in `/usr` not `/mnt/usr`. If you do `chroot /mnt` this will make it so that instead of looking in `/usr`, is looks in `/mnt/usr` as if that was actually `/usr`. When you run the chroot, it will alter your shell, and everything else you run in that shell will be "inside" `/mnt` (or whatever argument you use).

Chroot is a bit like running one Linux inside another. However, keep in mind that it's not like a true virtual machine, just some trickery with paths. You will still be using the kernel and device drivers of the host OS. Just the programs being executed will change. Depending on the complexity of the program you run, you may need to set up additional virtual directories like `/tmp` (chroot won't create these for you, it only covers real directories and files). This is why with just a basic chroot, GUI programs like Steam might not run, because they're not properly connected to the live OS's graphics server.

Arch Linux has a slightly extended version called [arch-chroot](https://man.archlinux.org/man/arch-chroot.8). This sets up some extra stuff so you don't have to do it manually. However, to use this, you'll probably need to boot from the Arch ISO, which has no graphics and might be difficult to use if you're not comfortable in the terminal. As a workaround, you can try the Manjaro ISO which has a similar `manjaro-chroot` but also comes with a GUI.

But if the program you are trying to run is specifically Steam or a Wine game, I think you might find that challenging. These rely on many specialized and obscure system features, and it's hard to provide all of those in a chroot, without booting into the regular OS. IMO running Steam/Wine inside chroot should be its own question, because it's very specialized and not the same as running simpler programs (CLI).

Normally what people do is boot from live USB, mount their hard drive, chroot into their normal system, use terminal commands to do fix it just enough to be bootable, then boot into the regular OS and troubleshoot further from there. You only really need to chroot from a live USB when the normal OS is unbootable. Once it boots, you don't need the live USB. As an aside, Arch Linux is even installed in the same way through a chroot, so you might find their [install guide](https://wiki.archlinux.org/title/Installation_guide) helpful. If you need a "working reference configuration", it's better to create a virtual machine inside your normal OS.

Good luck, and if you can narrow down to a more specific issue, try asking it as a separate question.