Why don't pushback tugs take the aircraft to the runway?
Way slower, more congested, more damage risk, many more expensive tugs required.  Taxiing under power is much faster and safer than towing in general (towing is a high risk operation and all sorts of bad things happen when tugs move large aircraft). It would be a mess on a taxiway with a line up waiting to take off with trucks and personnel mingling with ready to fly jets.  Generally, when you are flying you want the towing unit disconnected and gone ASAP. 

The only reason you have tugs at gates in the first place is because of the need to back up; they're just there for the pushback.  If someone designed a gate with a pass through design that didn't require backing up and allowed for at at least one engine running, you wouldn't have tugs there at all.