Does the level of a gym's badge affect the gameplay?

Leveling the Gym badge does have an affect on the game play. According to Niantic's post:

You can earn the opportunity to receive bonus items and increased rewards from Gyms by raising the level of your Gym Badge.

Having a higher Gym Badge level will yield more rewards and bonus items from Gyms. At the moment, this appears to be the only thing that the Gym Badges do that I can find.

My observation is that:

• a bronze gym badge grants you +1 bonus items
• a silver gym badge grants you +2 bonus items
• a gold gym badge grants you +3 bonus items

In addition, you get bonuses if if your team controls the gym:

• If your team controls the gym and you do not have a gold gym badge, you get +1 bonus items.
• If your team controls the gym and you do have a gold gym badge, then you get +2 bonus items (for a total of +5 bonus items, including the gold gym badge bonus).

In addition to receiving bonus items you also get more XP for spinning:

Enemy gym:

• badgeless: 25 XP
• bronze: 50 XP
• silver: 75 XP
• gold: 100 XP

Friendly gym:

• badgeless: 31 XP
• bronze: 63 XP
• silver: 94 XP
• gold: 125 XP