I want to know the FAA's reasoning behind issuing TFRs whenever a game is going on in a stadium. Is it just because of a large crowd of people? Do engine noises just bother people? Could an aircraft's wake blow a ball off course?
The 'stadium TFRs' were originally required by Congress after 9/11 to reassure people and encourage them to continue their normal activities without worrying about aircraft flying low overhead. And since then, Congress has repeatedly voted to maintain them; the FAA is just implementing them as directed.
There's more information in FAA Notice 8900.370:
Following the events of September 11, 2001, Congress enacted legislation that restricted civil aircraft operations over sporting events such as NASCAR, Major League Baseball (MLB), the National Football League (NFL), National Collegiate Athletic Association (NCAA) Division I Football, and other open-air events containing more than 30,000 people. These restrictions are implemented via a TFR. In 2004, Public Law (PL) 108-199, Consolidated Appropriations Act, Section 521 was modified to allow civil aircraft limited use of designated sporting event TFR airspace.
The TFRs are generally very unpopular with pilots and perhaps as a result, there have always been rumours about other reasons why they were created, or - even if you accept the original reasons - why Congress insists on maintaining them today. For example, the NFL benefits from not having 'unauthorized' aircraft taking pictures or towing banners over their games, when they could be selling those rights exclusively for a lot of money. So perhaps the NFL has pressured Congress to maintain the TFRs for 'security' reasons, while the real motive is making more money. The same comments and thinking are often applied to the 'Disney TFRs', by the way.
Having said all that, there are some arguments that 'security theater' - i.e. feel-good measures that provide no actual security - can have a positive impact on people's comfort and confidence, so it's not a black and white issue. But that whole topic belongs on another site.
All aircraft operations; including parachute jumping, unmanned aircraft and remote controlled aircraft, are prohibited within a 3NMR up to and including 3000ft AGL of any stadium having a seating capacity of 30,000 or more people ...
The 30,000 number indicates that the purpose of the TFR is related to the size of the gathering of people. The FAA doesn't issue TFRs for noise like that, and aircraft wake is not going to affect a sports ball in any meaningful way.
There are exceptions to the TFR for flights with specific purposes, or for any other flight that successfully obtains a waiver.