Why don't the districts better utilize outside resources? Because they're not allowed to.
Remember, the capital is continuously doing everything they can to oppress the remaining twelve districts. The capital specifically wants the citizens of the twelve districts to have just enough hope to continue to survive and produce goods and entertainment (by way of the Hunger Games) for them.
To this end, each district is surrounded by fences, and the perimeters are patrolled by armed guards sent by, and loyal to the capital. This means that citizens may not venture beyond the district as defined by the Capital, nor may they utilize any additional resources beyond the borders.
There is some additional discussion regarding the breaking of laws, and why people don't do it here.
Regarding the locations of the reapings themselves, these take place within each district. The citizens of each district typically never leave their districts unless they have been chosen for the Hunger Games. The first book is quite clear that the Reaping for District 12 takes place in their own public area.
At one o’clock, we head for the square. Attendance is mandatory unless you are on death’s door. This evening, officials will come around and check to see if this is the case. If not, you’ll be imprisoned. It’s too bad, really, that they hold the reaping in the square—one of the few places in District 12 that can be pleasant.
It also discusses the fact that the square doesn't have enough room for their own citizens, and that the Reaping is televised
The space gets tighter, more claustrophobic as people arrive. The square’s quite large, but not enough to hold District 12’s population of about eight thousand. Latecomers are directed to the adjacent streets, where they can watch the event on screens as it’s televised live by the state.
It goes on to confirm that the broadcast is seen all across Panem.
The mayor looks distressed. Since all of this is being televised, right now District 12 is the laughingstock of Panem, and he knows it.