We have driven to all parks but MK for EE mornings. It will be much faster to catch the bus to MK, as the bus will take you directly there, whereas with driving, you have to park, catch a tram to the monorail, then wait for the monorail to start (doesn't start until EE morning actually starts) and line up to get in.
We had no problems driving to the other parks, however, and never have had to take a tram to the entrance. Do be careful to distinguish your rental car from others, as the parking lots are not well numbered (especially MGM), and we have wasted precious time looking for that rental car that looked like everyone elses... out in the hot sun when we were ready to switch parks.
Also, from the Swan, it might be faster to walk to Epcot, and enter from the International Gateway... they do let guests in there for EE morning, too, and the line is always minimal.
The boats don't start until 7:30, but they are a nice alternative to driving or the busses, as well!