Aircraft are indeed designed to be pushed back by the nose wheel, and not pulled along by it.
Having said that, very large aircraft are picked up by the nose wheel and shuffled around Frankfurt all the time; even when empty, they're not that light.
If there was a big commercial incentive, then a "lift the main gear" tug could be used to trundle about to and from the runway. However, it's not likely to be worth the bother unless there's a very long taxi or queue. With fuel consumption on the ground no more than 10% of that in the air, it takes a lot of time to pay for a new class of tugs. Also, if there's a problem starting, then you're stuck miles from the gate in a line of aircraft in need of help.