I'm not sure why it would be silly to withdraw 100 riyals to pay for a 10 riyal item, unless you absolutely never intend to visit Qatar again or you have some insider information that the Saudi/UAE blockade of Qatar and the resulting pressure on the offshore value of the QAR has prompted the Qatar Central Bank to abolish the use of cash.
I still haven't worked out if any of the banks offer free ATM access to any of my cards. But I think maybe the threat of fees is one of the reasons people don't like to get small amounts from machines. Left over cash isn't such a great concern for me: I hang on to it until I'm in a friendly cash environment, like Bangkok, here the exchange window will hoover up just about anything at reasonable, commission-free rates.
Did you park any hand luggage at the airport? I can't see anyone lugging even a small bag around in hat heat.
No idea what frequency the bus operates, but i hope you didn't have spend too much time hanging around. I love the idea of you being the only passenger, but i think there's a message in there somewhere ....