So, this isn't going to be a popular opinion, but oh well.
My team at the beginning of last season placed me as the Engie. I was a demo/pyro main before hand, although I obviously had time under all classes, but I began to focus very heavily on Engie when I was selected for it. Before, I thought the wrangler and Mini were kind of annoying, but as I actually focused on the class, I realized something fairly simple.
I ask a question: In Highlander, what is the Engie supposed to do on offense?
Now you can quickly say "build stuff, duh!" But after the teles are down and the Dispenser is up at a choke, what then? You could build an L3 in the back to maybe catch flanks, but you become useless for the frontlines. I dare you to try and get an L3 up on the frontlines, or even just an L1, against a HL team that has even a remote idea of what its doing. Besides that, all you have is the shotgun. You're a slow, fat scout, and with less damage output.
After practicing and focusing on the class, I realize Minis and the wrangler are absolutely needed. Minis are designed for the very purpose of going into the front line in the middle of battle. If you remove either the full health spawn or the health regen, the mini will die 9 times out of 10 without firing a single shot. Once anyone sees a mini being deployed, its focused, and without BOTH the health and regen, the mini dies in seconds and becomes utterly useless.
The wrangler is situational. Defending first on Badwater comes to mind. Without the wrangler, the L3 above the tunnel exit becomes an easy target. You can't return fire to snipers, soldiers, demoman charged sticky shots, anything that can fire from the hill, and they will, and do spam the gun from the hill. Without the wrangler, it is an easy and quick kill. Its shield is necessary because it is ALWAYS spammed and focused, and its range is needed to help cover the needed areas. Wranglers are not perfect weapons, they make the engie completely defenseless himself, and easy pray to a spy, sniper or bombing soldier. Once the engie is killed, the sentry is idle for plenty of time to kill it without it returning to firing mode.
All the talk of "ban the wrangler and mini!" are unhelpful, because in the end, annoying != OP. It may be annoying, but so are sniper headshots and spy backstabs, but no one is looking to ban those. Without them, the engie is heavily gimped as an offensive class, and becomes near worthless on payload blu. The Engineer is the only class in TF2 where nearly all of the unlocks are viable, and they are mostly all needed depending on the situation.
EDIT: All that said, I do believe the auto-aim on the Wrangler should be removed, it feels unnecessary, and honestly is a problem for both the attacker and attackee. No one should be a fan of control being taken from them. The engie should have the skill to aim the weapon to use it. Its shitty for those being attacked, and is shitty for the user as it makes picking different targets from a distance more of a pain and less accurate.