I can confirm that that was the issue - apparently "minecraft.net" was blacklisted on my modem which was why I couldn't get skins to work, despite me being able to access all other aspects of Minecraft.net. Since I wasn't aware of this, I wasn't sure if it was an issue with the game client or as Galaxy_2Alex said, it may have been blocked (which was the case). Sorry that this was a false bug report, but thank you for helping solve my problem!
I can confirm that that was the issue - apparently "minecraft.net" was blacklisted on my modem which was why I couldn't get skins to work, despite me being able to access all other aspects of Minecraft.net. Since I wasn't aware of this, I wasn't sure if it was an issue with the game client or as Galaxy_2Alex said, it may have been blocked (which was the case). Sorry that this was a false bug report, but thank you for helping solve my problem!