Hi there! I've pushed a 3 year old thru Disney with an umbrella stroller and even she was too heavy. I'm only 5'5", but found the position of the handles to be terrible, and any time there were ruts it would jam in completely. It definitely wasn't comfortable enough for her to nap in - something it would have been fabulous for her to do.
I would second the thrift store/craigslist/used stroller idea if you have a very limited budget. There are some great strollers out there on Craigslist so you might find a gem.
We almost always fly Southwest, but gatechecking a stroller has never been a problem. In fact, the last trip, we used the stroller as a luggage trolley (probably not the best idea, but it worked) until we checked all the luggage, then gate checked it. I have a hard time believing there would be any fees for gate-checking a stroller - they would have to charge fees for wheelchairs too.
I just got back from a trip with 2 7-year olds, a 10-year old, and a 6 month old... we were worried about one of the 7-year olds because he has NO stamina, but he kept right up with the other kids, no problem.
Good luck!