How do you access your passwords on a computer that
doesn't have Keepass installed?
As far as people not using it, that's a social problem that would need to be addressed by company policy. The role of sysadmins in this is providing the tools to allow best practices, and to encourage their adoption.
That's an unreasonable demand. You can't anticipate which computer you're going to want to check your email on, for example. Now this might be a good idea for just work-related passwords (as they only need to be used on work computers), but the problem there is that for non-web applications, you're probably going to have to copy/paste the password from Keypass.
> use an OS that already has something similar installed
I don't think I actually need to explain this. There are plenty of reasons why using a non-Windows operating system is not an option for many companies.
There's also the issue of installing Keypass itself. I see it has MSI packages available, which I know sysadmins at big companies like. There might be some other technical requirements for the password-storage software itself.
I've been carrying around KeePass on an USB flashdrive for about 3 years now and haven't yet encountered a problem running it on random Windows machines. Deploying it is just a matter of copying 4 files to the target machine really.