there was a proposal for getting something by running a full node, without tapping or anything, and i remember that user that were runnign a node via tor were getting payed not long time ago
but what you suggest doesn't make, sense why incorporate a faucet when you can just go there on freebit.co and resolve your catpcha and get your peanuts in bitcoin?
and btw the tiemout is there to ensure that you don't spam with your bot to solve as many captcha as you want and get more than what the faucet can handle...
The new Bitcoin is rightly introduced by people who contribute to the network. How difficult it is to run a full node shouldn't change too much because hard drive sizes and bandwidth increase just as fast as the amount of them that you need to run it.
Freebitco.in is only running a free service because they can get ad revenue from giving out the dust payments. Bitcoin Core wouldn't, unless they make it really annoying for everyone. There's no point having dust payments in a full node, it wouldn't really legitimise running one or make people think it's a better idea, and it would be unclear why they would actually give out the satoshi.