In order to fix the hole in the hot air balloon, MacGyver places the map as a patch on the outside and duct tapes it at the four corners. For air escaping a balloon it would be much better to place the patch on the inside of the balloon and let the air pressure reduce the air leak.
When MacGyver climbs up to fix the hole in the airborne air balloon, a taut tethering rope can be clearly seen attached to the basket.
In addition to the other Goof already here, another mistake is that there appears to be only *one* hole in the balloon. Presumably there should be two holes, one for the bullet's entrance and another one for its exit. And it's hard to believe that only one soldier managed to hit the balloon.
The search plane had a US registration N4226C which would not be used or in South American countries. The Stinson 108 would not have the range to travel there from the US.