"Take Me Out To The Ballgame" may be a bit "cheesy", but so is much of the popular music of the late 19th and early 20th centuries that is in the public domain, at least to our "modern" taste. Besides, were the "Star Spangled Banner", "Oh Canada", and "Amazing Grace" really that "cheesy"?
It's also an appropriate piece for this time of the year, at least for us in the U.S. True, it's not quite fair for those outside the U.S. We often forget that there are other cultures and traditions in other countries, but not out of any malice.
Jason attempts to be magnanimous when making his choice of songs for the contest, choosing pieces with relatively simple melody lines and that are easy to learn. This is so as many people as possible have the opportunity to participate. Many of the classical pieces in the public domain are long, rather complicated, and are better with accompanyment, not making them conducive to being learned and somewhat mastered in only a couple of weeks. We are not all professional musicians or child prodigies.