I dont see where it states the visitor gets to use their own game balls...
Section 2
BALL SUPPLY
Each team will make 12 primary balls available for testing by the Referee two hours and 15 minutes prior to the starting time of the game to meet League requirements. The home team will also make 12 backup balls available for testing in all stadiums. In addition, the visitors, at their discretion, may bring 12 backup balls to be tested by the Referee for games held in outdoor stadiums. For all games, eight new footballs, sealed in a special box and shipped by the manufacturer to
the Referee, will be opened in the officials’ locker room two hours and 15 minutes prior to the starting time of the game.
These balls are to be specially marked by the Referee and used exclusively for the kicking game.
In the event a home team ball does not conform to specifications, or its supply is exhausted, the Referee shall secure a proper ball from the visitors and, failing that, use the best available ball. Any such circumstances must be reported to the Commissioner.
In case of rain or a wet, muddy, or slippery field, a playable ball shall be used at the request of t
he offensive team’s center.
The Game Clock shall not stop for such action (unless undue delay occurs).
Note: It is the responsibility of the home team to furnish playable balls at all times by attendants from either side of the playing
field.
OFFICIAL NFL
'An underinflated or deflated ball would be easier to grip, which would have been advantageous in the pouring rain Sunday night.'