This card says: place a satellite here, move 1 unit from its reserve to the board or from the board to its reserve. In a youtube video, the players were using this to basically recruit the "dead" soldiers and interceptors - moving them from the holding boxes in the middle of the "board" to the respective reserves. Is this allowed? It certainly makes the game massively easier - it is like five or six extra income each turn!

No, this is definitely not allowed - I can't find it in the rules/FAQ, but, as you observed, it makes the game massively easier, effectively makes the option to spend credits on recruitment obsolete, and removes almost all the cost of losing recruitable units - it actually makes moderate losses there less painful than losing Scientists/Satellites.

The obvious purpose of Mission Control is as a safety valve - letting you pull back units to avoid a budget overrun, or adjust unit assignments to react to a late threat. It also lets you pull soldiers off the Mission if it wasn't completed last round. It's not there to let you resurrect all your dead units every round.