What would the world be like without Shakespeare or the Olympics? While medical care and education are perhaps the highest causes to which money can be contributed, the arts and athletics are in some ways just as valuable. So, to ask the government to not extend support to these two areas could be just as detrimental to the welfare of society as the lack of sufficient health care and education.
The Romans believed in “mens sana in corpore sano”. In short, that there is indeed a correlation between a healthy body and healthy mind. When we are healthy, we feel better and so are more likely to be productive academically. Plus, the practice of sports can also teach us the very same discipline we need for our studies. Why even a brisk walk or watching an exciting athletic match can refresh the mind for greater work. Moreover, sports can serve to create healthier bodies, which in turn would serve as a form of “preventive medicine” thereby cutting down on medical costs. In the same vein, the arts are known to induce a sense of well-being in performers and audience alike, reducing mental problems and their associated physical manifestations and again, medical costs. As such, building theatres and stadiums, which spur interest in the arts and sports respectively, actually would be practically identical to spending money on medical care and education!
Now I realize the question specifically addresses the building of theatres and sports stadiums by government. Some people would contend athletics and the arts can still be enjoyed and practiced without such constructions. I would hazard that these buildings stand as the altars to the sports and arts worlds, inspiring would-be athletes and performers. Without such venues where would sports spectators and music aficionados be able to enjoy these events? Now some other people would also say business could support their construction, but we know the avarice of business could very likely jeopardize the lofty spirit of athletics and the arts just as well. Therefore, allowing government to retain some say in the matter on behalf of the people would, I believe, be in the best interest of the people.
In short, as long as the government does no go overboard in its expenditures for these buildings and uses such venues for the benefit of all, then, as mentioned above, people will benefit in terms of both medical care and education as well. This is not to say the government should neglect medical care and education, but rather to think of this not as an either-or choice but as a win-win situation for all.