I don't want to save my passwords on my computer. So I have to log in with password each time I use Steam. Most of the time I just want to log in and play.
With Steam Authentificator I have to type in my password and additionaly (!!) search my mobile, enter the code, and can then play.
When I save the login-password, I automatically saves the authentification status as well.
Then please always authenticate a computer or ask for it. The auth has nothing to do with the usage of the account to play the games. And also it has nothing to do with the password.
Since the password and the auth is saved, now a program could get both out of the cache. Great!

It's ok for me to use an authentification when I want to use the market or trade. It's then ok to ask for it and for example safe this information for that session. Why do I need a authentification just for playing?
In my bank-account I can log in with my password. Only when I want to do bank jobs I need to do an authentificaton with special hardware. Why not do the same here in Steam?

Ok, now my market/trade will take longer without authentifcator.
Maybe I will use a second account in the future for all of this. Then I have the advantages of both worlds.