Yes, the solo queue ranked system grades you based on your ability to play with randoms who may or may not actually be able to play their chosen position.
And yes, in general you will have to be Gold material to carry a Bronze team and probably somewhere in high Platinum-low Diamond to carry a Silver team consistently.
People like to claim otherwise, but most of those people don't realize that they were only carrying because their team was at least marginally competent. It is extremely difficult for one person to do the job of five regardless of how good they are.
That being said, if you are indeed better than the competition, you will eventually climb out. It will just take a long time and be a largely unrewarding and frustrating experience (one not at all helped by the existence of the promotional series).
Personally, I would suggest trying a champion other than Azir for the time being. He can be strong, certainly, but at the moment his weaknesses are similarly large and not at all helped by his numerous bugs and glitches. For instance, a huge portion of his potential is lost against anyone who plays semi-intelligently, and his combos take so long to execute that it is difficult to react to unexpected situations (or to surprise your enemies).
Second. As a midlaner you MUST roam. If your bot laners are having problems with feeding, chances are that means their enemies are pushed up to the turret all day and would be excellent targets for ganks. Don't worry about wards too much, chances are good that even if they have a ward down they will not notice you coming anyway. Not only can you set yourself up for success by getting assists and kills, you give your struggling bot lane some much-needed breathing room and time to catch up on farm. Even a completely failed gank will give your team space because the enemy will be nervous about your potential presence. So shove your lane out and get your ass bot lane. Helping your team is far more important than losing a few CS.
Third. Call out objectives when the opportunity arises. The enemy bot laners died to a gank? Get everyone to dragon, pronto. They're not going to have the damage output to contest you. Throw some wards down and keep an eye out for their jungler trying to steal it. And as far as objective priority goes, remember that turrets are always worth more than Baron. Your team will likely disagree with this assessment, but the simple fact is that turrets taken are a step toward victory. The first team to lose an inhibitor will (in a vast majority of cases) lose the game. Being the first team to secure Baron means very little by comparison. You go for Baron when you can't go for turrets, i.e. you only killed two of their team and can't push, but still hold the advantage.
Fourth. Once you get an inhibitor, leave that lane and don't come back to it until their inhibitor is about to respawn. This should be fairly obvious, but use the super minions to split the enemy forces up and give you an advantage in other lanes. Move around and take everything you can, and remember that if they all come to stop you that the super minions are going to wreck mayhem on their nexus turrets for you.
I hope this helps you out. Good luck, sir. =)
